No....not even an American Warmblood.
IF and only IF the horse is registered and APPROVED AWS or AWR can it be called an American Warmblood.
Canadian warmbloods are simular in that you can have what is truly considered warmblood but in the auxilliary division you can have anything including gaited horses. Draft crosses are discouraged but do exist.
The OPers horse is a TB cross or a Lipp cross. Get it registered something and only then can it be called anything different.
Shay-la has a "Canadian Warmblood" - a Clydesdale X Thoroughbred! Funny though, she really has no problem telling people she's a Clydie cross.
Just tell people she's a Lippizan cross.