The stat is a Braeburn Premier Series 5200.
The manual says that it is not required for battery only operation. I would assume that it could be a "power-stealing" type, or it can run without it.
How long do these batteries typically last in a thermostat? Can lithiums be used in it? I have had good luck with the Energizer Lithiums, they last forever in other devices around the house.
If they last around a year, then it really isn't an issue for me running it off of them. If I am going to have to replace the batteries every 3 months, then it is an issue for me.
The quote they gave me was for the complete end to end installation, including wiring if needed. Since they had 4 wires, they decided they didn't need the extra wires - Is that decision one that they should make or should I make that call? They were prepared to run new wires during the original install.