A cloned quarter horse is, genetically, nothing but a quarter horse. And it's the quarter horse association. I don't see grounds for objection.
If this matters to them, I suggest they rename the organization to the Quarter Horse Breeder's Association, since I think there's room to exclude cloning from the definition of breeding if they want to do so.
Personally, I find quarter horses gorgeous animals and favor keeping them around. Given the size of the genetic pool, I'm not sure whether cloning is a good thing or a bad thing in the long run. The right answer undoubtedly is that it depends on how it's used. Which may be the single best reason for bringing the clones into the group; "better to have them on the inside pissing out than on the outside pissing in."