Upgrades for that P?
mmmm, let me think!
I like your J Bass neck idea, but once you've got a rosewood board on that (and given that red body) I'd go for a pearloid pickguard.
A high mass bridge is always popular (and I think I prefer the aesthetic, even if it doesn't make any tonal impact).
The new neck will want new tuners (obviously); get yourself a big file to roll the fret edges, and sand the back of the neck to get rid of all the gloss.
But then there are the "go big or go home" options...
add a neck humbucker*, which means adding a third knob and drilling for a side jack...
*a nice wide range neck humbucker is so much cooler than adding a jazz at the bridge (yawn) and the CuNiFe sounds better (IMHO) than the EBO mudbucker Billy Sheehan used, but looks just as cool.