I think you are describing my bass. Is it the one on the L/H section of the Bass Gallery? Nearly 40 years ago I bought what must have been a 77 or 78 P Bass new, but customised it out of sight. It was a 'Friday afternoon' Fender bass and not much good, so I sold it. This one (which is also very close to the S/N) above, is VERY heavy, but sounds amazing. I could be playing in the Stranglers!
I feel like I've got my youth back (but not my back!)