Would the t-40 be able to produce a precision tone and then be more versatile on top of it?
I used a Fender Precision for 2 or 3 years in the 1970's, then replaced that with a Gibson Ripper, then replaced that with a couple of Peavey T-40's. At the time I was convinced that each new bass was an "upgrade" offering better sonic versatility and finer construction.
Admittedly, I have not played a T-40 since the early 1980's, but iirc the T-40 would still offer more sonic versatility than a P Bass
BUT
Three things to bear in mind:
1) the sound of a good P Bass is iconic, and you may find yourself in a situation where Close Only Counts With Horseshoes & Hand Grenades...the T-40 can get close, but I was never quite able to get that throaty formant that pervades a Precision's tone.
2) some folks judge with their eyes, not their ears. These folks always seem to be in positions of authority, like the MD of the pit orchestra or the producer of your band's forthcoming album. They won't know what a T-40 is, but they will know it's not a P Bass. They might want a P Bass. (Or think they want a P Bass.) It happens.
3) and this to me is the most important consideration: I find a P Bass is way more comfortable than a T-40. It's almost always lighter, the tummy cut conforms to my torso better, the fingerboard has a more hand-friendly radius (or rather, it
has a radius, unlike the T-40), in general a P is just a more comfortable bass to wear for long sessions than a T-40.
So if I had to do it all over again -- and I couldn't have both basses -- I'd stick with the Precision Bass.
Just my 2¢