Precission Neck reshaping

Hi All,
I purchased a new fender precision back in 1980 sight unseen and by the serial number I think that it's a 1978 build.
I have done a bit of work on it over the last 25 years including brass nut,stripped and refinished,fitting EMG active P-J pickups,new bridge,scratch plate,knobs.
One thing that I have never been happy with is the feel of the neck.I don't know if it's just me or what but it seems too thick.I am currently playing a Curbow and a short scale Hamer for my stage basses but would love to switch back to the Fender.You may ask why the hell did I buy it in the first place if I have done so many mods,the answer is simple.I was playing in a country rock band at the time and the image required was a P bass.(young and stupid)
To cut a long story short now is it possible to reshape the neck of a P bass?I have built a few basses over the years and I own a joinery so I have the skills to do it but I am not sure how far I can go before I find the truss rod or weeken the neck.
Any help would be much appreciated.
Cheers,Tomo