I'll throw another vote out for a Squier. Got a CV Jazz earlier this year (70s mind you, not 60s), and am loving it. Stock pickups are actually pretty decent, possibly good enough as a backup bass. I saw this as becoming my main bass for gigging covers, so given the use it has, and will continue to get, swapping the pickups out made sense. So far the bass has proven super robust; just got through a couple outdoor shows on days where we had all 4 seasons within the span of the gig, at a time where we're going from hot and humid to cool and dry, and the neck hasn't moved in the slightest! I was seriously expecting it to, so am really quite pleasantly surprised by how robust this thing has been straight out of the gate.
Almost went with Lollars, but ended up with Fralins, and I am LOVING them! If you go the swap route, be sure to give them a look too.
Don't worry about the P-heavy emphasis when playing Motown; I've gotten by using this J in a very traditional/purist Motown project, and they were all loving the tone. You can rock it for sure!