I can only think of two occasions that I have read where Batman fought Captain America and it was during that Marvel vs DC crossover of 1996. I liked the was it was done....respectful of both characters! I mean Batman is the best hand-to-hand combatant of the DC Universe and Captain America is the best in the Marvel Universe. Neither have superpowers so it was all about martial arts skill. Batman won because of luck and he admitted that. We all know that in real life, from our point of view, chance always plays a part in what ever happens. Therefore I was happy with that fight the most, compared to the others showcased in the limited series. I'm not really sure who would win if I had wrote the story. You can read the Comic Book Universe Battles for a take on Captain America vs Batman.
The other occasion is the long awaited JLA/Avenger crossover. I got it at this years' Wondercon! It's a great story. One of the best parts of it is where Captain America and Batman fought...feeling each other out and decided that they were too evenly matched and that their time should be better spent figuring out what was really going on and who their true enemy was.
As for the mugen battle for this post, I pitted Infinity Mugen Team's Batman against Infinity Mugen Team's Captain America and then Spiderbat's Batman versus Infinity Mugen Team's Captain America with each fight being computer controlled. Here is the fight.