I think the best bike for the job would be the bike you are used to. Bike polo has been popularized by bike messengers who ride track bikes all day, that's why they do bike polo on those bikes. I don't know if you've ever played, but I think it would be difficult on a BMX, standing up and pedaling one handed. Most polo matches have a "no dab" rule, too. The wheelbase on my track bike is 37" so the turning radius is about the same as a BMX w/a 21' tt.
Same with doing tricks, it grew out of bike messengers skidding, trackstanding, and hopping curbs as part of their daily riding. I think that is the misconception, that the "fixie" thing was something that hipsters (posers) started, but riding track bikes in the street is something that bike messengers made popular, and many of them used to skate or BMX and many still do. Hipsters (posers) caught wind of it, and blew it up into to what it is now, which is sort of a parody of itself.
A major difference between these two "cultures" is that fixed gear riders cover long distances by bike, which a track bike is much better at than a BMX, and many BMXers put their bike in a car to get to a spot. I am "guilty" of this as well, so I am not criticising.
FBM is a bunch of "Fat Bald Men" like myself, and we can't always fold up and fit on a BMX, and as you get older covering some miles on a track bike keeps your beer belly in check, but you still like to bust some style, so in those respects, a track bike is the best bike for the job.