Thanks! Based on all your much appreciated input, it's sounding more and more like most of my problems should be solved by losing the GFCI breaker and replacing all the receptacles with dedicated GFI's. I'll let you know how it works out.

Just so you know, I did what we talked about. I replaced all the outdoor receptacles with GFCIs and replaced the two-pole GFCI breaker with a pair of normal non-GFCI single-pole breakers. Plus I installed a dedicated non-AFCI circuit for the heat tape. That was more than a week ago, and everything's been working fine ever since. Those trough de-icers are great! It's SOOOOO nice not to have to break up ice several times a day and fish it out with my bare hands! And the horses really seem to appreciate the warmer water.

Thanks for your advice!