You can do it! If you're having difficulty getting enough words out, I recommend dropping by the IRC and asking for a sprint - usually there's someone around who'll be willing to do one!

Bingo fills should be created only for the bingo challenge, not for any other formal fandom challenge (except the tiny RB, this year). So it shouldn't be something you're planning to submit to the Big Bang, no. The BB also has a rule against using fic/art that you've submitted for other challenges, too, I believe. But hey, you can do it! And even if you fall a bit short, there'll still be a lot of fic created by trying :D