Sure you could keep your jazz neck (assuming pocket size is the same for Squier jazz and p bass) and buy a squier p bass body off ebay for $100 and some nice pickups for $50 and a custom pickguard and all electronics for $100 and a new headstock logo that says precision for $25 and a do it yourself headstock refin for $50 and a bridge for another $25. When all is said and done you'll have sunk about $350 into it not including your time and you'll have a p bass sounding and looking bass with a jazz width neck. If you want a p bass neck add about $100 to the build.
Or you could just go with one of these short scale killers that will run you less than $300 out the door:
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Or either one of these that will run you less than $400 out the door:
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I've tried the mustang and have one on order, one of the best vintage p bass tones I've heard in a long time (and I have an all original 1968 p bass). That short scale is really cool, very ergonomic and lightning fast. I can run riffs on the mustang with the short scale neck that I just can't pull off on my vintage p bass with the full scale neck.