In that case, and this is very generalised as you can change the tone of all bass models, a Rick might work well.
Going from most bottom end to most top end I'd put their core sound in the order of P, J then Rick. So if a J sounded clearer than the P, then a Rick might well be what you're looking for.
But as I hinted at above, depending on strings and EQ any bass can sound different. Stick flats on a J or Rick and use the neck pickup and it'll sound deeper than a P played with rounds and a plectrum.
In all honesty a good old P should cut through most bands if EQ'd right.