Where is this "Cap always plays fair" coming from?
Cap does fight dirty, especially if his opponent fights dirty as well. If he didn't, Cap would never have survived the war.
I agree with the "smarter" bit, and the belt gives Batman more options, but Cap's speed and reflexes are so above a normal humans that he can dodge bullets at point blank range; something Batman has never been able to do. And I'd say the shield is much better suited for a one-on-one punch out like this.
Also, the fact that Cap doesn't rely on weapons (not to mention his "throw my shield in just such a way that it'll ricochet of random crap and go right back to me") would make me think that he's even better at using his environment then Batman.