Here is an update for everyone.

We tried running the aerator on a timer but the stopping and starting of the aerator caused condensation in the line to freeze and cause issues, so we are just running it full time now. Our ice is more than thick enough at this point in the winter, so having it run full time does not seem to be an issue. I was fishing about 30 feet or so from the edge of the open area and the ice was 14" thick.

I have not even seen a brown trout while I've been fishing. I've only seen and caught rainbows. I have no idea where the browns go in the winter but they become next to impossible to locate or catch. The rainbows seem very healthy and have lots of isopods in their stomachs which appear to be keeping them well fed. Any mixing of the water that the aerator is doing does not seem to be making the rainbows any less aggressive. I don't know how the (unwanted) bass are fairing though.

Thanks again for all of the advice.