Bay City will only go up the property line, then it is the home owners responsibility.
I ran backhoes for years in the winter, breaking frost is terrible for the hoe. I've had the hoe lift boom snap in half from a stress break, that was a 2 year old tractor. Get/rent a ground heater and get rid of the frost as much as possible. Then a standard bucket can dig. Otherwise find a frost pick attachment as suggested, that is still rough on the backhoe however.