I just did that monologue and song for a concert for my choir, so I'm familiar with it. And I'm a humongous Bat Boy fan, so this is a fun thread for me.
I'd recommend to choose a song from something more contemporary just because Bat Boy isn't similar at all musically to anything in the Fantasticks, really.
However, for anyone doing this monologue and song or one of them, I HIGHLY recomned NOT doing it in the bland and boring way most people tend to do it. Look at the words--she's not an en genue who is naive, she's a naive girl who is very very different. When I heard the CD recording of the song I was bored to tears. Do the character justice if you're going to do her at all.
I agree with the recomendation of doing a monologue from a play, and depending on what part you want, try for something serious that shows you can act well.
Go listen to the soundtrack, whether you buy it or l isten to the clips on the official Bat Boy site (come on, you should have been there by now to find out about the show some more), and decide based on those what you think fits best.
http://www.freeiPods.com/?r=13865158 please =D