Will a custom pick up get you that tone? Not likely. (But you’ll likely get a great tone nonetheless).
For example. years back when I bought a 64 P bass I tried it up against 6 others (pre CBS). All were good, and totally different. 2 really stood out, but the others were still good. None of them were ‘better’ than any modern bass.
The feel of the bass is whole different thing. I’ve always felt there’s something so special about the wood used in the very early 60s, even student model Musicmaster and a Duo Sonic guitars I had from 1960 and 1961 were outstanding in feel and fullness of tone. The stack knobs and other Ps I’ve played were great too. That doesn’t mean I’m right about that particular period, just that I worked in the vintage trade for a good while and that was my experience.
And yeah, it’s a crime not to play them.