I spent the weekend recording above an unheated garage in Houlton Maine where it's about 12 degrees during the day, 0 at night. He had a space heater and the gear for warmth. I feel REALLY bad you're getting cold in the Carolinas :-)
The "Vornado" brand of forced air heaters are super quite. The good ones run about $100, and they are worth it. They have a thermostatically controlled and infinitely variable fan and heater element. Thy are very quiet at full power, but once the underside of the desk stabilizes, the Vornado will dial back the power and fan speed, and become virtually silent.
A small cat works quite well, though you may want a couple of them in case one gets sick of staying in the same spot and needs to be replaced.
Unfortunately the purr can add to the room noise, however if you add sound deadening material under the desk...................
I use an electric heater that doesn't have a fan. Not sure the brand offhand, it's just a heater coil with a round metal mirror (shaped like a satellite dish) that radiates the heat. You could even track with it on