Yes, don't look for me for tips on CBA style - I haven't a clue! I was playing a bit of the CBA at the weekend; the thumb is definitely on the fingerboard edge. Sticking it behind limits movement somewhat, so not great if you're up and down an octave or so - on the edge of the fingerboard gives more movement. Flash players can play with the thumb rather than sticking it anywhere; that's optional though. and a step too far for me. My understanding, from the RIP accordion forum, is that thumb playing is neither encouraged, nor discouraged by CBA teachers - it's purely personal choice.
Taught CBA players do tend to be able to play everything on 3 rows, using the 3 fingering scales rather than switching to the middle 3 and inner 3 rows to achieve the various keys - in which case, the fourth row is simply a helper row, used to make some fingering patterns a bit easier. So you can view a 4 row as a 3 row+accidentals... sort of.
Cheers,
Clive