Probably depends a lot on how quiet the heater is, and whether the hose is long enough the horse can be kept away from it. I’ve used heat-on-demand wall mounted water heaters with natural gas in England, and they hissed quite a bit. Also there might be fire safety issues from the barn’s point of view, if you end up storing propane at the barn.
As far as washing a horse in winter, even if you use warm water in the wash rack, you still end up with a horse that is wet all over and cold. I’d be disinclined to do a full body wash even with warm water in cooler temperatures. We do have hot tap water and I will sponge out mucky spots, then towel dry. For that you could bring a big thermos full of scalding water to mix with cold water in a bucket to make warm.