Light colors reflect light, dark colors absorb it. Putting black around the eyes would absorb the light rather than reflect it into his eyes. I have been told tattoos are somewhat painful, so the healing time might be upsetting for him. I would suggest getting a full-face fly mask that is made out of a material that will prevent the sun from shining through and burning your horse's face. Putting regular eyeliner on a horse is probably not an intirely good idea. It could work, but investing in a good flymask is probably a better thing.
As to why it isn't acceptable to have war paint on in some shows is the same reason you couldn't bleach a paint's hair to make it look better. Even if it is for health reasons, generally altering the animal's appearance is frowned upon if not against the rules. If you are going to show then you could just ride without anything and put it back on in the end. But with regular riding, it might be a good idea to see if there is any sort of bonnet or mask you could give him.