I guess a nice mustang would do it too
That's exactly why I said "allbreed". It shouldn't be limited to Lusitanos, Andalusians and Hispano-Arabs, IMO. Any horse built right should be capable of doing it, and you mentioned lower level....definitely there allbreed.
It gives more people the possibility to do it. If you limit it to the Iberian breeds only, you create an "elite" image. Unless that is what you want, of course