I've seen pics of working cowboys in the late 1880s-early 1900s wearing caps similar to the military muskrat caps. In the earliest pics, the caps looked to have a "peak" or "point" like the early military caps. None of the pics were clear enough to really identify the caps for certain.
That said, period accounts seem to mention wrapping a scarf or similar over the top of the cowboy hat and tying it under the chin. the brim of the hat folds down like ear flaps.
I've worn both in unpleasant weather (sub-zero, with winds >30 mph). The muskrat hat is warmer, brim could be a little longer to shade the eyes. Both call for a scarf or strip of blanketing wrapped around the neck to cover that unpleasantly-exposed spot at the base of the neck.
BTW, in that weather--not unusual for the High Plains--the winds are going to cut through your mackinaw like an ice knife. A duster over the top will help a lot, as will knit gloves worn under the horsehide mittens.