Should I go to commercial insurance or social security for reimbursement after discharge?

You go to social security for reimbursement first, but ask social security to issue you a separate bill, and keep a copy of all the fees, receipts, discharge summary and other materials you give to social security for yourself. After the social security reimbursement, take the social security split sheet and all copies before going to the commercial insurance declaration. General insurance companies need a separate form and a copy.

Further reading: How to buy insurance, which is good, and teach you how to avoid these "pits" of insurance.