In Reaper I would have the one chord track then use a send to another VSTi track and transpose the midi input to another octave for different VSTi's.
I think there was a transpose input just added to BB23 ?
But you could have one on each track at a different octave, you would just need Biab to keep track of what tracks they are on and the octave so they can be updated with chord sheet changes.
But to have options to do it whatever way suites the user is good.