I wouldn't worry too much. The lambs will be drinking little and often, not taking in a large amount of milk at a time like they would if fed say 4 times a day. It's highly unlikely to freeze inside an insulated stable at night, assuming the bucket is hung out of direct line of the open front, and assuming also that it is a plastic bucket. Plastic is an excellent insulator (unlike metal) and if you put a lid on, and even insulate it, (as suggested), then the temperature would have to drop to below about minus 10 deg C
in the building before the milk will freeze.
How do I know? Years of experience of how cold it has to be for our automatic, open topped, exposed to the elements plastic cattle drinkers to freeze.
Worse case scenario - if we did have (highly unlikely) arctic conditions and the milk froze overnight - the lambs would still be alive and waiting for their next feed in the morning.
However, it is possible to lower the freezing point by adding a small amount of glycerine. This is a constituent of twin lamb remedy so will merely increase the energy value of the milk as well as stop it freezing.