"I'm guessing it must be possible to play 'cross melodeon' in the same way that people play 'cross harp', and, I am hazy about all this, but I think it might bring in the mixolydian."

Exactly. Dorian is also available very easily.

"Maybe your idea only works, as you will be aware, if we exclude the ionian (ie major scale) from our category of modes."

Yes, I can't really be bothered calling major 'Ionian'.

"when a melodeon player hits a chord with the left hand buttons that to my ears clashes with the melody being played with the right hand buttons (and the chords on the instruments don't seem to have been chosen with the full range of modes in mind)"

This is very much a part of Cajun music, where they play one-row melodeons mostly on the pull, and don't have the chords available to provide a 'musically-correct' accompaniment.

"However, according to Roud/Bishop, many 'folk songs' are in ionian. So it looks as if Walter Pardon and Roud/Bishop disagree here."

The alternative modes sound more exotic, and hence perhaps older when compared to major tunes which are the most common in Music Hall and other more modern styles. Which of course is why folk revivalists throughout the last century or so have been fascinated by them.