If some has a better idea for the HVAC please post it
What will the sides be aginst? I know yo said it's by your bar but what about all the other sides? If it is not aginst any outside exposed walls I would not worry about piping in any air. The surrounding room will temper the room well. When we are installing systems at work we never install supply runs in interior rooms b/c it is surrounded by tempered rooms. Also when the A/C kicks on it will be pumping in dehumidified air. Your A/C systems tries to pull all the humidity out of the air to keep it cool. It can only get it to 45%-50% on a good system. So in the middle of summer the humidifier would be running non stop. Also you will need motorized dampers to use with that zone controler and an additional stat ran into the humidor.

Are you making the humidor a zone all by it's self? If so your furnace will most likely not work to well if at all when only the humidor needs heated. You would save alot of money and headaches by not zoning that room.