I'm not sure how well the low wattage birdbath heater will do outside in a run. I saw some birdbath and small koi pond water heaters on line that were 200-250 watts - that size should work for fine you but, I do suggest you insulate it well so it isn't so costly to run. As far as outside in subzero temps - A 1500 watt stock tank heater will keep a 150 - 300 gal outside water tank ice free however, they are not cheap to run.
My electric bill goes up by $50 per heater each winter month and that is in insulated & covered stock tanks.
If the water in the bucket is warm even a small ice tip on the nipple will easily break when the chicken bumps it.