I think you are way overshooting from a starting point without references. My advice is go find an American Pro and spend the next six months with it before deciding to either keep it, modify it, or exchange it. A vintage or custom shop bass doesn't have any magical properties that make them better basses. Vintage is expensive due to scarcity, and custom shop is expensive due to the people and process involved in building them.Definitely a Precision but there are huge distinctions between a genuine '61-4 in this context against any Custom shop one is paying $5000 for what one will never get toward the other. Can't get Brazilian rosewood and even if one could it might not be of the maturity Fenders originals had. I read today Fender on a small number of Precision's used mahogany bodies in 1963 but ash and poplar. Poplar! I guess it comes down to which of either sounds the best to my ears. A 1963 Precision or a 2018 Custom shop Precision? The woods on the prior were mature to begin with. That has changed, I think, with all makers. I am saying, too, without having any or either in my hands to feel and hear differences, how can I know? In terms of other manufacturers who make what are essentially P's and J's, I can't know until I might meet at a place with vintage and newer and various amps that replicate what I wish to sound like for myself. Then there is a purpose for all my folderol. This has all become an intellectual pursuit for things I'll probably never have nor can afford. Lately I relax to my old record collection and get very excited about them all which happen to have been produced in the 50's through the late 60's. Hate Disco but some of those guys played awesome on those records but not enough to get me jumping up and down. I'd like to have a sound that makes others jump up and down. That's the bottom line for me. We'll see....
If you truly want to treat this as an intellectual pursuit, the first thing to do is establish a baseline.