Well... as sort of a non-update, the tech came for an inspection last week and suspects some switch thing between the AC and heater is dead. They are going to follow up with the manufacturer first to see if anything is warranty-able, but even then, the labour is going to kill me.
Crap can I just go back to electric baseboard heating? I can safely say that as far as I can remember (since my teenage years), I have never had an electric baseboard heater failing / breaking on me or require (any?) maintenance.