What can I do to stop nursing facility from becoming the representative payee authorized over mom's social security?

Guest is correct. I will assume that this NH will be permanent. In my instance, Mom was running out of money for her AL. Mom private paid for two months before Medicaid took over. I had her SS payee changed to the NH and her small pension. I had to sign allowing this as POA. NH handled the paperwork. I would think this is permanent so see no problem in doing this this way. As said, once she is in a home all her money goes to her care especially if Medicaid is involved or maybe in the future. Upkeep of home and car will have to be taken care of by family or sold. I chose selling. My nephew was living with her and he pays the utilities. I chose to not pay taxes so that will be a lean on the house when sold. As will be Medicaid.

Who signed the admission paperwork? If your parent is competent, they can change the Authorized Representative with Social Security. Are you the Power of Attorney? Social Security doesn't recognize power of attorney and requires that a person be appointed Authorized Representative for payment. How did it get changed? More info will get you better answers as you've now got 2 posts going.
By the way, if your parent has applied for Medicaid, all of her social security besides a small personal needs allowance will have to be paid to the nursing home. There will be NO money available to keep a separate family residence or pay credit card bills any more.