Sounds to me like your amp and cab are perfectly fine, and you just need some effects pedals and/or lessons.
Perhaps a compressor pedal that also adds some tonal coloration, like the Diamond, might be just the ticket. And a good teacher can give you advice on how to attack the strings so your notes sound punchy and cut through the mix. When was your last lesson?
Also, what kind of bass are you playing, and what kind of strings? Maybe just a new set of strings will give you what you want.
Or he doesnt like the tone out of the Rumble. I hate the Rumble because of this.
OP - to answer your question...
Different heads will sound different. Ampeg sounds nothing like the Rumble. Mesa sounda different than both of those, as does the Mark.
The only solution is to try as many out as possible.
My Ric or Jazz plugged straight onto the Mesa D800 has plenty of grit when dialed in for that. No pedals needed.