Ask the DMV clerk for a copy of their regulation

or policy proving that their rejection is based on that policy (no GC, no license).

If they can't produce one. Try to argue:
"If you take away my right to drive based on what you think personally, not based on law/regulation, then you violated my constitutional right to travel and you discriminate me because of my status as an alien, get your supervisor, otherwise, I will hold the whole department liable for my loss".
See if it works.

If they have a written policy, then you cannot fight unless u want to file a suit for constitutional violation against the state.

Disclaimer: FYI only, not a legal advice.