I think Batman would win. Although we need to determine what kind of fight it would be. Would it be no-holds-barred to the death, or would it be an honorable fight to determine who was better?
If its the first, Bats might have some trouble winning since he doesnāt kill, but if worse came to worse, he could strap a huge magnet to the Batmobile, get Wolvie stuck to it somehow, and then just drive it off a pier into the drink till Wolvie drowned. This fight would be tough for Bats if it was close combat, because of the claws, but I doubt Batman would stick around to go hand-to-hand in that case. Heād probably tag Wolverine with some sort of tracking gas/spray/nanotech (like he did against the Predator), then get out of there and come back with his magneto-batmobile.
If its the second, Wolverine might actually have a better chance. Going hand to hand with no claws or gadgets (being an honorable fight after all), Wolverine does still have his healing factor to protect him, and his adamantium to act as a weapon, even if it just makes his punches that much harder and more weighted. Although Batman has probably studied in-depth all the pressure points on a persons body and might be able to disable Wolverine that way. Iād still lean towards Batman pulling out a win somehow, maybe with a chokehold, since Logan still needs oxygen. But Bats would be hurting after taking a couple of adamantium weighted punches. Wolverine does have his samurai training, but Batman has studied martial arts all over the world. And I think Wolverine just relies on his claws a bit too much. He can come up with a great plan and follow through on it occasionally, but as Mike Tyson said āEveryone has a plan till they get punched in the face.ā Batman would probably tag Wolvie till he lost control and his animal side came out, then win since Wolvie would no longer be able to strategize at that point.
As far as Wolverines abilities, Wolverines senses can be disabled (anyone remember when Sabertooth shoved wasabi up Logans nose to take out his sense of smell?). Iām sure Batman could come up with some sort of sonic emitter to take out the hearing, and then just kill the lights since Bats has some night vision lenses built into his cowl. And does anyone not think that Bruce Wayne isntā good enought in chemistry to come up with some knockout dart that would take even Logan down for a little bit. Wolverine does have his claws and healing factor, but even a big enough explosion will still knock him out for a while (see the recent fight between him and Deadpool, although he did mean to lose that one). Plus Wolverines healing factor has been scaled back recently after his encounter with Lazaer, so heās no longer as indestructable as he once was (which he probably was relying on for a while, heck, he had his entire body torched off by Nitro and still came back). Bats has always been human and so he knows his limitations better.
All in all, whatever it would come down to, I would still lean towards a win by Batman, no matter how slim the margin.