IMO your issues have nothing to do with the SD2, but to eliminate that as a cause, I would try connecting MIDI from accordion directly to Tyros 2.

What Fran said, I would make sure the Tyros is set to "Fingered" mode, or possibly "Fingered on Bass." Your split point has to be set correctly to include the octave(s) of the accordion's chord buttons. Unfortunately you probably cannot change the inversions of the chords as they come out of the accordion. As I recall, on Stradella bass, many of the chords don't use the root at lowest position, because the accordion has separate bass buttons.
Another gotcha...some accordions output the bass and counterbass on a different channel.
I KNOW the Tyros's can be programmed to recognize bass and chords via separate MIDI receive channels. It's just a matter of getting everything on the correct channel. Unfortunately I can't give you step-by-step instructions, but Keep trying. Find a local musician who's knowledgeable about MIDI and they might help you.