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Sell or Keep?

As for the Rumble 15, I'd just keep it. It may come in handy for trips and such. Like, I took my Bassman 25 camping once. Playing bass under the stars by the fire was really cool! (Except my bass smelled like wood smoke later, LOL).
 
"Warm, thumping tone". Fender Rumble head's have a dip around 400hz which I think is killing the ability to create a "warm" tone for you. Putting a pre-amp in front of the Fender may help with this, however, that would be a waste of money.

Search used gear and graduate away from combo amps - get separate pieces. Louder and better tone are sure to follow. Getting another combo amp would be great for your bedroom tone, but it is basically a lateral move if you are planning on playing show eventually.

Trade your combo amp for another combo amp and a little money. Maybe an outright trade. craigslist.org is your friend! :bassist:
 
With a budget of $300 I would definitely look used. Should be at least 300 watts, and 15" speaker min. Ideally you would like to have a separate amp and cab, but that might blow your budget. Oh and a 410 cab would be ideal as well.

Brands?

Look at Acoustic, GK, Hartke or maybe even Peavy. Carvin is another one, but I have no personal experience with that brand.