The Jack Bass (Hohner) are really teriffic value for money and a really comfortable shape to wear. The tone tends to quite an organic, woody sound, but can do "modern" sounds with a tweak of the eq. I used to run mine passive.
The neck cross section is quite round and deep by modern standards, and narrower than a "P" at the nut, but good to play for long periods. And they never go out of tune, just like the B2As you've seen. The ones with humbuckers probably get nearer to the P tone, but the pickup placement isn't quite the same. That said, you do get more tonal flexibility, even with the passive models.
For what you can pick them up for 2nd hand, it's worth trying one, you won't lose money if you sell it on.
They do make nice fretlesses too if you whip the frets out.
Only caveat: check there is enough adjustment on the truss rod.