Bishop-Dabbaba: If a Bishop-Dabbabarider is a General, perhaps the BD could be a Major. (I was going to say Colonel, but that's taken by a Charging Knight/Rook.)
Rook-Alfil: I'm thinking about what would be a rotary counterpart to Major. Miner (as a pun on Minor) seems possible, but a tad trite. Farmer, perhaps?
Queen-Alfil-Dabbaba: Taking the initials, it could be a Quad.
Knight-Pawn: That's new information to me, and it will affect some of the names in my Dragon Wars variant. Thanks for that!
Commenting on some of Jean-Louis's suggestions: