The main problem with your dual-purpose idea is that anything decent at going down a drag strip will be too modified for any autocross class in which it'd potentially be competitive, and anything that fits in an autocross class where you stand even a chance probably will be pretty regular in the 1/4 mile.

I'm no autocrosser, but the lighter the car the easier it is to make it handle good. If it was me, I'd pick something relatively lightweight that I happen to like, and go have fun with it.