What sort of box is in question?
A friend of mine plays a CBA, using LH almost exclusively out of his LMH 3 voice box. It sounds quite nice for the sort of music he plays, but I would not regard that sound as an all-purpose machine - the lack of a M voice just makes this sound a little weak IMO.
So a different, pleasing sound, but not 'more boom'.
I've done work on a hohner presswood model, with the objective of getting more boom. New italian reeds, modified blocks and chambers, modified holes in soundboard, and a whole lot of tweaking to get the RIGHT tuning (not just going wet, but finding the right sound for that box), and some other details did the trick. That box cut right through any sessions I brought it to.
You say the MM option is just too weak for your style. Well, perhaps you need a different kind of box, then? You can't make a 2 voice MM, MH, LH, or LM box sound like a 4 voice cajun box or a 3 voice musette box....