I have a spare LUX digital thermostat I would like to use. Isn't there a way to hook it up to an appliance module and make it work that way?
No. The Appliance Module needs a PLC input from the powerline. If the LUX will run soley on batteries you could still go the Power Flash route.
If you want the direct route, which seems to be what you are asking for, you can use a 24V AC transformer (door bell transformer) and a 24V AC coil relay. You can get a 24V AC coil relay from any heating - cooling dealer (they are typically called "fan relays" in the HVAC community). The transformer powers the LUX and the LUX controls the relay through the Air Conditioning contacts. Cut the "hot lead of an extension cord (the hot is the non ribbed wire) and use the N.O. contacts of the relay to close the circuit.
I'm still trailing Brian...