WTH?
Anyway. Batman being Batman, he'll ditch the armor and therefore do much better. He'll also refuse to kill Wolverine, because he's Batman.
The omnipotent thread starter/dojo master will get pissed and send in Winter Soldier, who Batman will then also take down.
In other words, Batman wins, (with or without armor) but wont kill Logan.
Batman could not beat wolverine one on one that's why he has the armour. W/ the armour he would probably kill wolverine which is why winter soldier comes in and then since bucky has no healing factor the armour wouldn't really affect him so he would find it easier fighting batman and during the fight he would probably be able to damage the armour or use his electric charge to weaken batman which would in turn make the fight between him and wolverine easier.
Batman already defeats Wolverine without any armor.