I researched this idea and ended up with an insulated bucket instead.
Beet juice on roads works mostly by making the normal road salt sticky. It will work by itself for deicing roads if the concentration of the beet sugar is high enough. The higher the concentration of sugar, the lower the freeze point. A ten percent solution freezes at about 22 degrees F (-5 or -6 C).
I wasn't willing to give them that much sugar even if worked better than my alternatives.
At 22F, the water in my 3-quart vacuum-sealed ice bucket (made to chill wine) took over 12 hours to get a skim of ice on top. Well, during the day, when they disturbed the surface by their drinking that is. Since they don't drink after dark, I could have refilled their bucket once a day and given them all they wanted. I checked them twice a day anyway.
If you have a larger flock; it gets easier to use insulation.
The other reason I didn't do it is the sugar keeps the water fluid but it doesn't keep it warmer. I think water too far below freezing (pth, below the freezing point of tap water, that is) could damage their throat or mouth tissues when they drank it. I know they will eat really cold snow but they can't eat that as fast as they can drink.