There is no issue with just changing the P-pickup so long as that is the only pickup you are using on the bass. After the safari to get a new P-pickup to your liking, then you have several options with the J:
1) The J-pickup may or may not be in the same phase as the P-pickup, so you may or may not have to swap the wires on the J-pickup;
2) If you do get a high-quality P-pickup, you may want to change the J-pickup as well to something like a split-coil humbucker, also to keep the noise down and the tonal quality up.
3) Check the rest of the wiring first to make sure you have good quality pots and tone cap; what you describe as your tonal preference is typical vintage 250 kohm pots and a .047 tone cap.
4) Since you do have a P/J, as I do on a travel bass, I upgraded the pickups and wiring so I always have the option of the pickup combinations, even if I don't use it much.