I can't really say from experience, but here's what I have noticed.
I thought about using a cover over a BA bridge. I was going to get a BA bridge and then saw one on a Geddy sig. jazz bass and realized that BA bridge was MUCH smaller than I imagined it to be!
So, I think you could put a jazz style bridge cover over a BA bridge. The p style cover would most likely be much too small, though.
If you go for a jazz style cover (I think they are the best looking anyway... I like their beefiness over the smaller p covers), the generic off brand j covers you see everywhere are great quality. Nice thick steel and great chrome.
But... plug the bass in and play it before you get a cover. You might not like the BA bridge anyway. It seems like half the people who buy BA bridges love them, and the other half think they're garbage...
So, I say get a sense of the bass's sound first. If you think the bridge is funky, put a Fender bent steel bridge on, sell the BA, then you can pay for your cover and then some.
