I have a modified yourockguitar midi guitar, which is similar to your Yamaha.
Open chord changes aren't really practical as you've discovered. With midi guitar you have to adapt your playing, to the midi guitar and to the instrument you're triggering. For Em to Dm I would either play higher up the fretboard and transpose or not play the low E and A strings.
I'm surprised that the fishman lets the low strings ring out, are you definitely muting them when you change chords?
With all midi guitar recording I think you have to expect to do a little bit of cleaning up in the midi editor.
This might be useful for the fishman, I haven't used it personally, but it dumps open strings below a specified threshold:
http://adam.fulara.com/software#MidiDumper