Should I Play The Bassoon?

The more important question is whether you want to or not.  You should take her request as a compliment as it sounds like she believes you can do this.  I, as a fellow musician, would visit with your director more and find out exactly what she expects.  The bassoon is a completely different type of instrument.  Flute music is written on treble clef whereas the bassoon is written on bass clef.  I was once in your position when asked to play saxaphone after years of playing clarinet.  The difference there is that many of the fingerings are the same.  Good Luck and keep up the good work.  You should be proud of yourself.

A lot of it depends on whether or not you really want to. I don't think it'd hurt to try, it may even help give you a new appreciation and/or new ideas/ways of playing the flute. For example, my favorite electric bass player, john entwhistle, started his musical career as a french horn player. He used what he learned from that and applied it to bass and started to do things on bass others had never imagined before. He really opened up the role of bass in rock music ;)