Is the money already in the business account for the down payment? If so, the best thing to do is to let it season for over 30 to 60 days.
A clear operating agreement will also need to be structured for the LLC and since I read the goal being that the relative is not to sign on the loan, they need to be kept at less than a 20% partner.
You will have to go private money I believe on this one. Wish you the best of luck on your hunt.
Ask around to different lenders. I'm not sure on this but if the down payment is made as a capital contribution to the LLC that may work, the best thing to do is talk to multiple lenders
@Andrew Nolan If you have a JV agreement some lenders will accept down from family member. I think you just need to clearly explain where it came from and ask directly if lender is OK with it.