Wear lady pants and makeup, 'androgynous' look, but go to boys room and wear your name badge. I'd go along, try to speed up DL change and SSN.
Your employer is not really discriminating you, they feel that they must follow some policies according to your legal gender. They accept you as a trans and won't fire you when you change your documents.... You need money to keep the ball rolling, and there are many jobs where women have to follow appropriate dress code and not where whatever they feel like.
So, if I were in your shoes - I'd keep the job, continued to search for a better one, tried to dress androgynously and try to find reasonable balance, pushed for HRT and DL/SSN change. Good luck
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