Somehow the "International Generic Horse Association" doesn't sound very prestigious. I think I would do what GrayRay suggested and just give him a full name and a barn name.
I went through the same thing with my foal Zane. He's a QH/Fox Trotter cross. I call him Zane, but if I ever wanted him to sound more proper, I would call him Zane Gray. I thought about registering him with a pleasure gaited horse registry, if he shows gait, but then I thought, who the heck has ever heard of the gaited pleasure horse registry? It's not like I'm going to breed him or anything (he's a gelding). So why register with a registry that is not taken seriously? All they are doing is taking your money and giving you a piece of paper in return.
My Mustang John is another good example. The last owner just called him John, so that's what I call him too, but on some of his paperwork that came with him, such as health papers, his name is listed as John Henry.
So I don't think you need registration papers just to give your pony a real name. Just have fun with it.
At a show, no one is going to be none-the-wiser when his name is called out in the results if he is registered or not.