Hey Diogenes, best of luck to you. I'm currently a junior and I'm interest in law school after graduation. And I will be facing the same thing. I think that you should try to get the loan yourself if you have a good history, along with the common banks, you could try sallie Mae. I don't have experience getting student loans but I've done conventional and I go directly to my credit union and I just ask for a loan; they ask for my ID and social security number, they haven't asked to see my social security card.
Anyways, I wish you the best and I hope you get the best possible outcome on continuing your education.