Batman is one of the greatest superheroes ever. Even though he got no powers, he accomplished many feats such as being the only non-powered superhero in the Justice League, solving unsolvable crimes, and beating many people asses including Superman.
He is one of the greatest superheroes in Marvel (If not, the best.) He has adamantium claws and skeleton, healing factor, superhuman senses and superhuman condition and is well-trained since he fought in many wars.
Wolverine somehow entered the DC universe and he is asking random homeless people where is he? He gets angry and brutally kills them with his claws. Batman arrives on the scene, seeing Wolverine killing innocent people. They have all out brawl.
It would be a HELL of a fight, but Wolverine would go on because of his healing factor and Batman will get tired and then Wolverine would finish off Batman. But if Batman escapes and analyze the material Wolverine's claws are made out of, he could somehow developed a formula to recreate the adamantium alloy into the form of a weapon. Then he could come back and stealth attack Wolverine from behind, decapitate Wolverine's head and remove his head from close of Wolverine and wait 12 seconds and Wolverine dies. But it depends if when Batman jumps and is about to decapitate Wolverine's head, and Wolverine turns around quick and stabs Batman in the brain, neck, or heart or decapitate Batman's head, it is over.