The school that sponsors you J1 visa is responsible for everything you do. If you get a job, if you move, if you drop a class…that institution has to approve it all. Even if it is allowed, which I don’t think you’d be allowed to enroll at another school, the sponsoring school would have to approve it.
It might be possible for your sponsoring school to make an arrangement for you to take a class at another school, under its registration.