IMHO, the fatness is going to increase as you have less baked-in speaker tone, less baked in amp tone, and more control over EQ.
The cab is a given, so yes, you could spend $XXXX on a new tube head, but seriously, why not try a pedal that makes your Classic Vibe P go active? Much more EQ range that way.
A VT Bass, ToneHammer, or one of the Sansamps would be a good place to start, IMHO. This is a good way to deal with baked-in tone and the limits of the passive P-bass EQ.
If you have enough power, stick with it. If not, I would not send my bucks down the "tubes are the answer to every question" route. Find a powerful clean head that can support a wide range of cabs and pre-amps.
Also, you could do a pre amp or pedal as a preamp now, if you have an effects-receive that defeats the built-in preamp and your headroom is good.