Sell it and buy a reverse P bass.
I can't hear what you're hearing so can't say for sure, but maybe your just don't like the sound of a standard P bass.
A reverse P put the E/A pickup closer to the bridge to give slightly more attack and less boomy, with the D/G side nearer the neck to give a fuller, rounder sounding top end. The result gives a better balanced sound across all strings, and solves the woolly low end, thin top end inherent in a standard P.
This is of course subjective and open to debate but that's the theory and it might just fix your 'problem'.
Maybe try one out.