I'm not familiar with Dave Allen or his pickups but there's really no need to think boutique for a vintage tone. The original Fenders were hardly boutique , often wound to "looks about right" rather than exact number of turns or a set DCR.
The Roswell is a good pickup but it's not a vintage tone IMO. Those large slugs don't hit the spot in the way a normal slugs does. They also have a much higher wire count than the original single , Roswells are up in the 9k range while 7k would be more in keeping with your needs.
Sadly 7k ohm often means a lower output so the Bootstrap Palo Alto might be better. Just $35 and only a 3 week lead time. You have the option of black or white cloth around the coil.
Bootstrap Palo Alto for '51 P
First thing I would advise is get rid of the D'Addario roundwound strings. Old school single coil P-bass tone is either tapes or flats
Pots are just fine. You have a Log 250k for tone/ Lin 250k for vol. If you decide to swap the pots for 2 x 250k Log found in old Fenders ; don't buy full size CTS pots with USA thread size . They'll likely be a tight fit in the control cavity and the control plate would need reamed out to accept the shaft.
Cap is 0.068uf , pretty close to the original 0.05uf.
Regarding "baseball bat" necks. This term has confused me for years. I've owned 6 PB-50s, set up plenty more. I've never encountered one with a neck I'd describe as thick/chunky or a bat

Sure they're reasonably wide , it's a P-bass

I'd say you're on the money thinking Squier VM
TL

R
Change the strings to flats. Go for a pickup with smaller slugs and a lower DCR.
Final thing. Enjoy the bass