Not necessarily laws...but someone has to bring them over. Probably no one had any interest. Size isn't the most important consideration for the export of a breed to a new land.
But beware of using the word "Ainu" or "Inu" loosely since it's a term for their native population who live mainly on the Northern most island and who are actually caucasion. To link a name of a people along with a dog is considered a gross insult in Asia. The population is not well regarded by most Japanese. Make sure you're working with an accurate breed name and not derogatory term for someone.
That actually happened to someone I know who was looking for a special type of working dog breed in Korea. Some guy thought it would be a cruel joke to send him into a neighboring village asking this question. Almost started a riot :( :(
regards,
toodles