Given that you've gone to the trouble of getting a true Jazz bass dimensioned neck, I'd have thought that you are really beholden to construct a Jazz bass with it. Use the neck from the JB kit to make a project bass.

With you being in Florida, with large variations in temp and humidity, I'd be tempted to give it a couple of coats of whatever finish you normally use, and lemon oil the board if it's not maple. It probably won't affect it if you don't, but better safe than sorry.