I really dig the tone of the brown '77 USA ash body bass as well, by far the best sounding in that video in my opinion, though followed by the white '95 Japan basswood body bass, even if that too lack some top end bite in comparison.
It's like the brown one emphasis the whole frequency spectrum although slightly weighted towards a pleasant amount of punching mids and some high end bite and sizzle to top it off, while the others seems to cut off quite drastically at some point and instead emphasis the lower mids, making them slightly more rounded or soft sounding, but at the same time also slightly tame and boring, like a teeth less tired old lion or something.
If you are not using the volume and tone knobs anyway and never cut your tone or volume you might as well try to bypass all controls and caps and hook your pickup directly up to the output plug of the bass, this should open up the pickup considerably, mainly in the higher mids/treble frequency area.
Did so on my Ibanez Mikro equipped with a set of P/J EMG Geezer Butler Pickups, and it really did.
Prefer the tone I got now slightly over the tone I got before, since I kind of like my tone raw and slightly aggressive with a lot of punch and a fair deal of growl and bite as well.
Swapping strings might help too, cobalt or even steel strings instead of nickle wound should make your bass sound slightly brighter as well.
But as mentioned by RobbieK, it will still depend on how your bass sound acoustically.
What kind of wood is your P made of?
Finally there's the option of you just getting a bass with P/J configuration, to add some bite from the J pickup