Hey @Fons, welcome to the forum! I love that bass too - although a lot of people complained on that video that they couldn’t pick out the sound of the bass very well, and I’m guessing it was because of the classic dark tone of a P bass with flats. Oops.
I don’t know Squier’s P Basses personally, but I’d hazard a guess that if you find one you like in a music store and slap some LaBella flatwounds on it, it’ll sound close enough for your liking. Or any other P Bass, or even a PJ setup, like the Yamaha Broad Bass, or a Squier PJ.
One thing I’m learning over the years as I buy and sell gear - you never really know how much you’ll like a bass until you play it a bit. I bought a 5 string version of my Peavey Cirrus thinking I’d love it just as much, but turns out the tone just isn’t the same, even on the 4 strings they share. Seemed like a safe buy based on how similar they looked, but seems don’t make it so. (but seams do make it sew)
And Joe @Jazzbass19, I think Fons was referring to the sunburst P bass, not the white J bass.