To answer a couple of your questions ... Squier made what they call a Precision Special, it is a P bass body, with a P pup in the neck position and a Jazz pup in the bridge position ... neck is the same neck used on the Jazz Standard series, 1.5" at the nut ... perhaps my favorite profile of any neck, at any price ... they havent made these for a couple years, but they are still out there new old stock, and used all over the place for under $200 .. my personal opinion is that I prefer either a P or a J over a PJ, but some on here would consider it the best of both worlds ...
... as far as amps, decent on the cheap .. since you like Hartke and GK, there are decent (older used) heads quite inexpensively in both, look for the older Mosfet type 2000/3500's in Hartke, and the BL 600 or 400RB (preferred) in GK most of these are avail for $200 range or less (2000's and BL600's probably $125+range) and then add your $150-200 cab of choice ... Hartke cabs used are quite inexpensive ... just a few thougts that would have you spending less than your original post and with a lot better gear and resale at or near what you have into it ... JMHO
Edit to add: your very first comments were speaking of a MIM Standard P bass .. dont rule them out either, I am seeing some sell VERY inexpensively these days, and if you can pick up one of these for around $200-250, its tough to go wrong ...some on here will tell you that a Squier CV or VM will be better, I will say that the Squiers do seem to be more consistent, but a $200 MIM Standard P with a working truss rod is better than money in the bank ... again JMHO
