Number one, as Wolverine has said many times, "I'm the best there is at what I do. And what I do ain't pretty, bub."
That being said, Wolverine, without his adamantium skeleton looses to Batman? Why's that?
Because Wolverine is the best there is at what he does in the Marvel U. Batman competes in the DCU, and let's face it, to be an ordinary human being and survive in the DCU is pretty impressive. The DCU has all the power houses, the heroes and villians in it are far stronger, on average, than those in the Marvel U. Therefore, Bats has to be a whole lot better skill wise to hold his own.
Also this is non-unbreakable skeleton Wolverine, and as someone pointed out, it takes Wolverine a few hours or a couple of days to heal broken bones. Of his normal skeleton the only part that is more durable are his bone claws, every other bone in Wolverine's body, without adamantium is just as strong as the bones in any other healthy person. In this case Batman can and will break Wolverine's bones, putting him out of the fight for a few hours or a few days, that means Wolverine is helpess while Batman is fine. Because he could block and avoid Wolverine's without taking fatal wounds for long enough to break Wolvie's bones. That's just the way that goes down.
Now, if we're talking adamantium skeleton Wolverine, he wins, because then Batman will have a hell of a hard time inflicting damage on Wolverine that will keep him out of the fight more than a few minutes. Even with his gagdets. That's just the way of things.