I was on a pee quest last year. The problem with Fender is there is no consitency in neck shape and profile through out the years on any of their pee bass models. Body woods can vary too. So IMO; for the money, the Fender Nate Mendel signature pee bass is a prett good bargain. Not to mention consistent build quality on the model. I personally like the neck profile and nut width on this particular model. The only caveat on the Mendel pee is the pup. I'm not a big fan of the spb3 quarter pounder Fender installs. Install an emg gzr and you'll have yourself a very nice 70's reissue pee for pennies on the dollar. It's the best bargain out there IMO.
I agree 100%. The Nate Mendel is a very nice model with an ash body and terrific neck. I changed the pickups in mine to Aguilars as I do not like the 1/4 lbers. Mine also wears TI Flats. It has the sound you are looking for. Shop for a used one. Pull the Duncans and the bridge and sell them to get closer to your budget.
Another option is to build one. You could start with an MJT body and get any wood/color you want and then add a neck, pickups and hardware to your liking.