..., my boyfriend goes bananas if anything from the coop comes inside. ...
What do you guys (especially those who live in cold weather) do? Is there any evidence that I can throw at him to support washing the bowls indoors?
My dh doesn't want chickens at all. He graciously tolerates a LOT more mess than he wants; even doing the chicken chores when I'm gone. I am grateful he is supportive when he so much doesn't want to have them. I try to be supportive of how he wants to live. If I were you, I would find a way to wash them outside.
One thing is it should need cleaning less often as microbes grow much less quickly in colder temperatures. You may be able to let the cleaning go through the coldest spells and pick the warmest days to wash the bowls. Even in climates like yours and mine, this can solve the problem very late into the fall and from very early in the spring. Even in the coldest parts of winter, this helps.
It is possible to safely wash bowls outside in below freezing weather. It isn't that much different than some of the things I did for four to six hours twice a day when milking cows and related chores in the upper midwest (just to say this isn't just theoretical even though my dh tolerates washing in the kitchen sink)
Wearing latex or neoprene gloves makes that safer and more comfortable. Having lots of towels handy, helps, Take some really, really hot water to add to the bucket as the water cools. Scrub out as much as you can with stiff brushes or scrapers before using water. Doing it out of the wind is key. Wear a good hat and boots while you do it; keeping the rest of you good and warm will help your hands tolerate it. A waterproof apron helps, especially as you learn how to do it both quickly and without splashing (it can be hard to do both for some people). Setting up a place to do it can help (like a bench at a comfortable height - lower than counter tops, so water runs down off your fingers rather than down off your elbows.) Having a surface to catch splashes that you can shake or beat ice off of (like a small piece of heavy tarp) or switch out (like a box of kitty litter, maybe).
Pick your location carefully. Any water splashed or thrown out will be there for the rest of the winter so make sure it is out of the way. And won't run where you don't want it when it melts in the spring, even if it melts quickly.
If there is some above freezing place your boyfriend is okay with setting the bowls to thaw, even if it doesn't have a sink, can help... you can brush and wipe them mostly clean after they thaw, leaving much less wet part to do outside. This might not apply to you. I soak their food at least once every couple of days, so it would help in my set up.