We've had sub-zero weather, including windchills at -12`. As a Californian in the Midwest, we're kind of out of the loop on tricky winterization kinks.
My home was built 1901 and while comfortable, the ground floor water closet at north side of house likes to freeze the inlet line. The heater strip doesn't seem to do the trick. Adjacent sink with separate taps is fine. After a couple 10` days another line supplying a W/C in 2nd story might follow suit.
It's easy to set a slow drip on faucets...but water closets just don't have such a feature.
Until now.
I cut 3 narrow strips of cellophane and put one under each flapper valve, one down stairs , 2 up stairs. They're 1/2" wide, folded over once, so it's about .004 thick; a very small orifice. Too small to clog anything, it causes the level to drop so float will react and refill the tank. Seems to take about 30 minutes.
There are spare strips in each restroom, and since I also rotate which is being used, this may alleviate the problem.