I have never directly compared the Mustang i had to a P Bass, (I did have a '97 California Precision Special (CPS) for a while, years before getting the Mustang and I did not have them at the same time). The CPS is a P/J type with a Jazz Bass neck but I never really bonded with the sound of it, it was always kind of thin sounding to me, but the Mustang was fat, meaty, and could be somewhat grindy when desired. Mine had a Jazz Bass feel by the neck, very slim and thin, and I used TI Jazz Flats on it. The tone control was very effective and could help produce a very deep or mellow tone depending on how I played the bass.
I agree that active electronics are not a necessity to sounding good so long as the bass itself has good tone. Since you do want some variety, the suggestions for a hollowbody is a good one, I used to have a hollowbody Rickenbacker bass for 38 years, it was the first Rick I had ever bought (in 1973), but it is a 34" scale length and they are only available on the used market for significant $$$, these days. Can you deal with a "standard" size body (Gibson/Epiphone/Guild/Gretsch, etc.) or a smaller body like the various violin shaped ones? Personally, if I were to buy another Höfner, it would be a 500/2 Club Bass instead of the 500/1 "Beatle" bass, otherwise I'd go for a standard size one, probably a Guild of some sort.