Your forgetting some details

Many gothamites belive batman is a demon or a vampire which could very well be true given demons and vampires both are canmon

You're asking the wrong questions. Could L find Batman? Possibly. Would Batman already have a plan to stop L from finding him? The SECOND L googled Gotham the first time, is phone wouldnt stop ringing from his clients across the globe with emergency situations.

This is the type of crossover where the first half they try to outsmart and outwit each other as detectives, then end with a threat ( or a case ) that requires them to collaborate.

Yes, L would eventually be able to because Bane did when he was introduced. If someone smart enough to watch, he would figure it out.

L would figure out who Batman is fairly effortlessly.

Proving it would be harder but doable.

Catching him?
absolutely not.  What is L's win condition here? Because even if he can prove it's Bruce, I have no doubt Bruce would simply disappear.

I think L will know that Bruce is Batman, but no hard proof. Even if he does, the most he'll do is have one frank conversation with Bruce, and then leave him alone, knowing that he won.

9:45 "L would avoid using his own name."
Death Note would be over in an instant if he wasn't already doing this.

Better question is would Batman catch L?

Going by the title alone: yes, absolutely. Less intelligent people in the DC universes have figured out Batman's identity.

Whether or not he'd out Batman and put him and his loved ones in danger from every criminal, corrupt cop, and lawyer looking for a massive lawsuit payday in Gotham is another question.

Catch the Batman? Reasonably, no. However, if people like Bane & D.A.V.E can figure out that Bruce Wayne is Batman by either deducing it or frankly, using information available to anyone, I think L could figure it out to; given how smart he is. For giggles, I wrote a little scenario between the two. Enjoy!!

The air in the Batcave was dead quiet, save for the rhythmic, low-frequency hum of the supercomputer and the steady drip of moisture from the limestone ceiling.

Batman stepped out of the shadows near the garage bay, his cowl framing a scowl of absolute, razor-sharp focus. He didn't reach for a Batarang. He didn't need to. He was looking right at the center of his security grid, where a young, pale man with unruly black hair and dark circles under his eyes sat cross-legged on the ergonomic leather chair.

L was casually spinning a half-empty mug of heavily sugared black coffee by its handle, his bare feet tucked up against the edge of the seat. The massive monitors of the Batcomputer glowed against his face, displaying real-time schematics of Wayne Enterprises' orbital defense grid.

Batman’s cape billowed slightly as he closed the distance, his voice a low, gravelly growl that had rattled supervillains twice his size.

"You're sitting in my chair. On my network. Which means you bypassed biometric firewalls, military-grade encryption, and a perimeter defense system that can track a falling leaf." Batman's fist clenched. "Explain who you are, and why you aren't already paralyzed on the floor."

L didn't flinch. He didn't even look away from the monitor. Instead, he took a slow, deliberate sip of his coffee, set the mug down, and finally turned his dark, unblinking eyes toward the Dark Knight.

"It is a pleasure to finally meet you, Batman," L said, his voice a smooth, flat cadence. "Or should I say, Mr. Wayne? Though, given the echoes in this cavern, I suppose Master Wayne would have been appropriate if Alfred were currently upstairs."

Batman didn't move a muscle. "You have ten seconds."

"Ten seconds is plenty of time," L replied, holding up a single finger. "First, regarding how I 'legally' got here. I didn't hack your firewalls, nor did I break your locks. At approximately 3:14 AM, my associates filed a federal petition for an emergency administrative search warrant through the Gotham Circuit Court, citing anomalous, high-frequency energy spikes originating from the bedrock beneath Wayne Manor—spikes matching the unique particulate signature of a Wayne Enterprises military prototype missing from a secure R&D vault."

Batman's eyes narrowed. "A forged warrant. The GCPD would never sign off on an arbitrary raid of my estate."

"Not the GCPD," L corrected, a faint, almost imperceptible smirk touching his lips. "I bypassed local corruption entirely. I routed the emergency writ through a federal district judge in Metropolis, leveraging a clause regarding suspected unauthorized trafficking of Wayne-manufactured alien-alloy components. Furthermore, I included a clause for immediate execution due to 'imminent risk of evidence destruction.' When the federal marshals arrived at your front gate with a valid, signed writ, Alfred—wisely prioritizing your long-term legal standing over a shootout—let them in."

"And the marshals?" Batman demanded. "They wouldn't let anyone wander down into the cave unattended."

"True," L nodded, picking up a sugar cube and balancing it delicately on top of another. "Which is why, while the federal marshals were busy cataloging your grandfather's vintage car collection upstairs, I simply walked down the primary hydraulic freight elevator. It was left unlocked. Frankly, for a vigilante who values paranoia, your maintenance staff's operational security leaves a great deal to be desired."

The sugar tower collapsed. L didn't blink.

"As for how I deduced your identity," L continued, leaning forward, resting his chin on his knees, "the deduction itself was remarkably elementary. It took me less than forty-eight hours after arriving in Gotham."

Batman remained silent, waiting.

"It began with a statistical anomaly," L explained, his tone analytical and detached. "Bruce Wayne is a man infamous for public decadence, staggering wealth, erratic philanthropic attendance, and an aversion to any schedule that requires physical exertion before noon. Yet, a cross-reference of Bruce Wayne’s private flight logs, credit card transactions, and biometric check-ins at various galas revealed a recurring pattern: whenever a high-profile, physically demanding vigilante event occurred in the city—such as the breakout at Arkham Asylum three months ago—Mr. Wayne’s public footprint vanished for precisely the duration of the crisis."

L pointed a pale finger at the monitor, bringing up a split-screen video feed. On the left was footage of Batman grappling across a Gotham rooftop. On the right was high-speed thermal traffic camera footage of a sleek, custom-built black roadster leaving the perimeter of Wayne Manor at midnight.

"Coincidences are acceptable," L said softly. "Patterns are informative. But physical evidence is absolute. During your altercation with the Penguin's enforcers last Tuesday, a small fragment of your composite cowl armor was left embedded in a concrete pillar. My forensic team analyzed its molecular structure. It is a proprietary carbon-nanotube weave. I ran a query across global industrial databases to see who manufactured that exact batch of material. The buyer wasn't the United States military. It wasn't LexCorp. It was a subsidiary shell company owned entirely by Wayne Enterprises, shipped directly to a private hangar registered to your name."

L tilted his head, studying Batman's imposing, armored form from head to toe.

"Add to that your approximate height, your stance, the subtle asymmetrical hitch in your left stride—a healed micro-fracture in the tibia, likely sustained during a fall from a height of roughly forty feet—which matches an orthopedic x-ray filed under a pseudonymous emergency clinic visit in Zurich five years ago, when Mr. Wayne happened to be vacationing in Europe."

L folded his hands back around his knees.

"The conclusion was mathematically inescapable. Batman is Bruce Wayne. And Wayne Manor houses your base of operations."

The cavern fell back into a heavy, suffocating silence. Batman stared down at the young detective, processing the sheer, terrifying scope of the intellect sitting before him. No supernatural magic. No psychic tracking. Just cold, unyielding logic, backed by a global network that had unraveled a billionaire's life's work in two days.

Slowly, Batman's posture relaxed—not out of comfort, but out of calculation. His hand drifted down, not to a weapon, but to the console of the Batcomputer.

"You have the warrant," Batman said, his voice dropping an octave, carrying a warning weight. "You have your deduction. What happens now, L? Do you arrest me? Do you broadcast it to the world?"

L picked up his coffee mug, taking a slow sip.

"Broadcast it? Heavens, no," L murmured, his dark eyes sparkling with a sudden, sharp amusement. "I have no interest in locking up Gotham's primary deterrent against organized crime. Besides..." L gestured toward a secondary monitor displaying a flashing red alert regarding a complex financial cyber-attack originating from international waters. "...it appears your city has far more pressing problems. And quite frankly, watching you work is much more entertaining from the inside."

L looked back up at the Dark Knight, offering a rare, lopsided smile.

"Now, Mr. Wayne... do you happen to have any strawberries? This coffee is dreadfully bitter without them."

I want this fanfiction on my desk by Monday

Woukd love to see a mixed scenario of L trying to figure out kira yoshikage and bring him to justice, one with the jojos at that part and one without

If L tries to expose Batman he would get jumped by everyone in Bruce’s roque gallery

This video is heavily reliant on which era Batman L is chasing, the newer interpretations simply wouldn't get caught. His plot armor is too thick, even for fan theories to intervene. His prep time has prep time and just asking Superman, Martian Manhunter or Nightwing to sub in would make things difficult.

Not to mention, Flash is a literal Ex Machina, if Bruce thought L was a credible threat, he wouldn't mind using Justice League talent or resources.

Something like Batman during his first years, where he actually used a gun and tossed people off rooftops would probably be easier, but L would have to be wary of Batman leaving a trail of sugar cubes or hitting him with an insulin injection... yeah early era Batman is effing ridiculous.

One thing adaptations miss and is less obvious in comics is Bruce's philanthropy. Bruce wayne has publically funded the Wayne Foundation, other charities, development projects, and political campaigns for the benefit of the city. This would probably help L in identifying Bruce Wayne's protective relationship with gotham, which could help L identify Bruce as Batman.

Lets be honest, one of the biggest flaws with Batman as a series is that anyone with an IQ over 95 would be able to point to Bruce Wayne as a likely candidate to be Batman if they give the notion a bit of thought.

Tim Drake figured it out at age 9 simply from a photo of Robin (Dick Grayson) performing a unique acrobatic move. Could L catch Tim Drake?

The answer is yes, because no matter how Batman launders his money, it’s impossible for it to become truly untraceable because of the sheer mountain necessary for all his projects and equipment. L’s ability to trace dark money would be higher than Batman’s ability to make money dark.

This is litterly a Naruto and Sasuke dynamic. One had a family the other didnt

I still can't believe with how smart L is, he is not part of the smartest anime people manga crossover and Light is