Now I do have a P with flats, my 1966 sunburst that I've had since 1973, wearing a set of 12 year old LaBella flats. Old school thump with a vengeance.
Last year I picked up this pair of American Standard Fives. Been experimenting with strings. I decided to leave the Fender roundwounds on the P. The meatyness of the pickup seems to gime me plenty of thump but also some zing and growl with the roundwounds. But I wasn't really happy with the sound of the J. It seemed really weak, thin and nasal to me (I've been playing Stingrays, fours and fives, and my '81 G&L for the last few years).
So I decided to try flats on the J, just the standard Fender medium flatwound set. What a difference!
Meaty, thick, round, and yet articulate. Made the J my go-to bass for my last three gigs. Sits in the mix great, blends with the bass drum, made me enjoy playing the J more than I ever have.
So I know it goes against common wisdom, but that's what working for me, rounds on the P and flats on the J.