I have the same problem! My horse is also that really bright coppery red color that is BEAUTIFUL but very hard to match. I would go with natural colors or maybe a natural base color with just a little bit of blue, turquoise, or something of that sort. When you get him all bathed and primped for a show he is going to be so shiny that you don't want to have too many other busy colors contrasting with him.
You could also try something like Mayatex's metallic reversibles if you wanted somthing with a little more "zing". They are a little more flashy without being busy.
http://mayatex.com/store/show/1104M_San_Juan_Metallic_Reversible
It's tricky with that color horse! when you figure it out put up a picture!