What the mortgage company wants is to have you personally liable for the mortgage. A C Corporation's protection is too great than what they want.
My suggestion is Option 2, that you transfer the property to your own name; after closing, transfer it back to the C Corp like what you did the last time.
Do not do LLC. It is the same reason that C corporation has more protection.
Tax event. Basically, make sure that the C Corporation pays for all the expenses and enjoys all the benefits. Then, transferring the property back and forth is just for you personally to assume the loan liability, and to be personally liable for the debt. If the check is paid to you, the 10,000 is loan and is not income. However, in the future, if the company is going to pay for the debt, this become your benefit. Therefore, make sure that you transfer the entire 10,000 to the corporation.
There is a doctrine called "substance over form". The only economic effect and business purpose is to obtain the loan refinance. The transfer property back and forth should not be a taxable event.
Please feel free to follow up.
Fiona Chen, MPA, Ph.D., CPA, ABV, CFF, CITP