I would be tempted to put an Arduino at both ends, use two of the wires to provide 24VAC to the thermostat and the other two for two-way communication between the Arduinos. Then the thermostat Arduino can use commands to the other Arduino to turn the heater, cooler, and fan on an off.
Better yet, put the batteries and charger by the furnace and send 5V up the wires to the thermostat.