I have one toilet that freezes when it gets cold and wont flush. Of Course its the one I use in the Middle of the night.So there I am on my knees cleaning up the mess at 3 in the morning. I havent found a plumber yet who can tell me why this happens as drains dont usually freeze.
By the way the heat tape works with electricity, If the power goes off, No heat.
My drains freeze too, I can cope with no water for a day or too but no drain sucks big time.
The cold wind blows, up the stack, and that toilet is tapped directly into the stack.
That's my amateur opinion.
My amateur fix would be to set the float on that toilet so that the water level is low enough not to overflow with a flush, typical "contents" and through the fill cycle.
At least you could get back to bed sooner, and worry about the full, but un-overflowed ceramic fixture, in the daylight hours.