Doug Yeo uses a shallow cup mouthpiece built on the same rim as his bass trombone mouthpiece. I suppose you could call it a Yamaha 62A. A number of other players I know use shallow cups on a 50 or 51 sized rim. For me, the smaller cups work better. Pretty much every bass trumpet I have played has been pretty stuffy below about D in the staff. Larger mouthpieces (in my experience) don't do anything to help this.
If you get a bass trumpet I recommend you learn to read F parts. 90% of the work I get on bass trumpet is last minute sub calls to cover horn parts in quintets. If you are looking for a piece to show it off in a recital, try the Neruda Trumpet Concerto down an octave. It lies really nicely in the sweet spot of most bass trumpets.
Finally, a story. I do not know for certain if this is true, but it's one of those stories that ought to be true, even if it isn't. Back in the Reiner era Renold Schilke was playing bass trumpet during a rehearsal. Schilke was using a large trumpet mouthpiece which apparently irked the Maestro. Reiner had him play a passage, shook his head and explained that to get a proper sound on bass trumpet you needed to use a trombone mouthpiece. He then told Schilke to show up the next day with a trombone mouthpiece.
Schilke went to his shop after rehearsal, took a trombone blank and made a copy of his trumpet mouthpiece inside it. The next day Reiner stopped the orchestra after the same passage. He pointed to Schilke and said, "See, now that is the true bass trumpet sound."



