Math problem. Energy usage is on the yellow Energy Star sticker. Capital cost amortized over life of unit + expected operating cost. Competing heat sources for small spaces are portable propane heater, inset wood or pellet stove. Connecting to your whole house system is probably a nonstarter. For cooling, wall unit, fan (hand or electric) or a kiddie pool with a block of ice. Insulation of the new space is most important.
Fair warning: I had some anesthetized dental work today so I'm still a little foggy. Proceed at your own risk. May pick a fight with before it wears off to given him more of a chance.