Yep, I was afraid that your two biggest hurdles were no grid power and LOTS of cattle to water. That is a tough combination.

I can't remember, are your cattle fenced off from your big watershed lake, or do they have access to the water in the summer?

I am asking because I think a separate cattle pond built AT GRADE may be your most idiot proof solution for a water supply that can survive bitter cold conditions.

You could excavate a small cattle pond just beyond your existing lake fence line, and get the bottom below your "normal" winter water elevation. Run a 6" pipeline to your lake so there is a constant supply of water to to the cattle pond to replace their drinking requirement. You can then install a solar panel and battery system to run an aerator and keep a section of that pond open for cattle access.

I suspect you also have existing small cattle ponds scattered around your property that perhaps get too low during the winter? If one of those is "near" your big lake, you could use a similar system as I described above. On one of the "warm" Kansas days that occurs in a cold snap, you would need to fire up your gasoline powered trash pump and move water from the lake to your winter cattle pond. A 2" trash pump (150 gpm) or a 3" trash pump (280 gpm) can move a lot of water.

You just need a drain valve at the bottom of your discharge line to the winter pond. Open the valve and drain the line after each use, and then a pipeline on the surface would still be usable during the winter.

I am throwing out crazy alternatives because constantly pumping water is a lot of HP for an off-grid electrical system. If you want to go that route, calculate your cattle water requirement PLUS some cushion for freeze prevention circulation and then size a centrifugal pump to match that flow. Post the HP of that pump on the forum, and some of the solar experts can estimate the solar panel and battery requirements to have that pump run for a week straight under a January sun. I believe that will require a very large (and expensive) system.

P.S. I also can't remember if you or some of your family lives full time out at your property? If so, then an air pump by the house and grid electricity, plus LOTS of air pipeline may be your cheapest and easiest solution?