When taking in hand shots do you try to place a horse in the
shade? I find that if I place a horse in the sun, it has dark
shadows in certain areas where the horse's body shades the sun.
Also some horses close down their eyes (squint). Black horses
in the shade seem to come out dull though.
All of the problems you are having can be corrected by using flash
while shooting out of doors. As a matter of fact, I personally
use flash at all shooting sessions, whether in or out of doors.
You have listed all the reasons for using flash, perfectly.
Let me know your response and the results if you decide to try fill flash.