US Lipizzan Society will register half Lippizans, if you are interested in jumping through all the hoops.
However, as is the case with most Baroque and Warmblood breeds, the North American registries are not considered as prestigious as the European ones. For Baroque breeds (like the Lipizzan) this means that a North American registered horse (who is not European registered) cannot ever produce a horse eligible to be European registered.
So IMO the $70 for DNA testing, other fees for registration, etc.. are going to cost more than what value they will add to the horse by registering him as a half breed in a "make-due" registry.
They say money doesn't buy happiness -- well happiness doesn't buy horses!