I think the fretted Mouse can pull off a gig that would be appropriate for a P Bass. It won't have the thump and growl of a P, but IMO it will work if you are willing to give up the exact P tone in favor of the short scale and ultra light weight.
Regarding fancy woods, etc., I again (and Rob agrees) don't hear a ton of difference between the wood combinations (I have a fretted Mouse with F hole, Koa top, rosewood board, and a fretless with no F hole, maple top, and the standard Algonco or whatever that wood is called fretboard. They sound very similar.
Rob's standard build for the Mouse (nice flamed maple top, alder, and the Algonco fretboard) is wonderful sounding and looking, and will save you a lot of money.
I have a clip of my fretted Mouse on my YouTube channel, which might give you an idea of the tone you can expect.
IMO and IME!