You'll only get 7-8 hours of operation with a 9V alkaline battery from 20" of light strips. I just tested a 3528 white strip 20" long and at 9.5 volts it drew 65ma, at 9V it was down to 55ma. The 9V alkaline is good for 500 milli-amp hours, and realistic you'll get more like 400 before they're pretty dim.
It'll work, but it's not going to be cheap feeding it batteries all the time.