Here's another thought.
I have never tried this, but I see no reason why it shouldn't work for the temps you experienced as long as you insulate the pipe.
Wrap the shortest Home Depot heat tape around the pipe (they are intended to be laid on the pipe, not wrapped) and power it with 12 volts. No, it won't get as warm (it will get about 1/10th as warm....maybe a little warmer since you wrapped it), but if you leave it on all night and your pipe is insulated, I suspect that you will be fine for temps into the single digits. And, if it doesn't work, then simply just power it with your inverter (120 volts) and use a timer to cycle it on/off.
