The ability to flip the polarity (phase) of the pickup relative to the mic probably won't matter too much in a gig situation; it will affect the sound when your mix pot is between pickup and mic, and I'd just suggest experimenting with the polarity when you build the thing and then leaving it in the configuration you prefer. Whatever you do, you'll never get perfect phase alignment between pickup and mic - the best you can do is find the sound you like best.
I'd suggest starting by deciding what mic or pickup you're going to use - that will have a large bearing on the design of your circuit, and on how resistant the instrument is to feedback.
I'd recommend using a standard PP3 size battery, as it'll be easy to source, change, carry spares, etc.