Squier P5. Stock pickup is fine. I recommend changing the tuners.
I had a VM P5 that I modded that should be below the budget with money left over to replace the tuners, pots and pickups. The neck was straight, balance was good. I even painted it Sherwood Green (the actual reason I bought it was to paint a white bass a vintage color so when it wore through it would look like a white undercoat) and only sold it to make room for more projects. I gigged with that bass, may make another in Lake Placid Blue.