I feel you. oh do I feel you. My Brisco boy is all white and he doens't know that there is such word as "clean."
In the winter, I don't show so I don't worry about it as much, but when I do get a chance, I usually like to use some form of cowboy magic. or I just rub him really really nicely with a rubber curry comb and go over twith a hard brush, then with a soft brush. Then when I am done riding I let him roll and it starts all over again the next time we ride.
Unless you are showing, I honestly wouldn't worry about it through the winter. During the summer when I get ready to show and begin practice again, I try to keep him clean as possible. Then A week before show I clean him and this year will blanker him because he an't stay clean.
That would be one of your best bets honestly if you really want to keep him clean, is to blanket him.
No matter what I do, even after a bath a week straight (not that i do) he won't stay clean.
🔫 Don't Tread On Me 🐍