Fender Am Pro P5 V-Mod pickup

While I absolutely love this bass, it wasn't easy for me to find the right strings for it and I don't know why. My 50s Mex RI P (upgraded with a Seymour Duncan sdp-1 vintage pu) has the perfect P tone for me, but the B-string is missing and it isn't nearly as smooth to play than the AM pro V. The neck and the frets are much better on my AM pro P5, too.
While both basses are strung with chromes in the same size it sounds different.
Would a pickup swap help to get me in the same ball park?

I uploaded a video with both basses, same strings (chromes, 45-100). Preamp is a Darkglass Vintage Ultra V2 with everything at noon. I chose that over a clean signal because that is the way I record and play live anyway.

 

any ideas for a replacement pickup?

 

Any opinions on that one? NP5F Fender