The battery would generally not even last a day, let alone an entire winter...
A small bird bath deicer is 50W, a decent heater is about 150W...
A typical car battery has a capacity of 45Amp/Hours...
If you were to run a 50W heater with a 45AH battery combined with an inverter (assuming real world loses, and battery drain curves) you would only get about 6 hours before the battery was dead...
If you were to run a 150W heater with a 45AH battery combined with an inverter (assuming real world loses, and battery drain curves) you would only get about 2 hours before the battery was dead...
That assumes the heater is running full time, if the heater is cycling on/off this would be the approximate on cycle...
You can get longer run times with a decent deep cell battery, but most of them are around 120-150AH so you are still under a single day run time...
And you would need a very sizable ($$$) solar array to keep the battery charged in the short winter days a that level of drain...