If I have more prizes remaining than my opponent and I play Black Belt, but then I KO one or more of my opponent's Pokemon before I attack, would I still be able to use the effect of Black Belt if I'm no longer behind on prizes?
Yes, you can. The requirement of being behind on prizes is just for playing the card, and once it's in play you can use its effect even if you are no longer behind.
Source: PUI Rules Team (2011-01-27)