If he can not buy you out then he can not buy half the house and pay you off. Can you buy himj out? If you can buy him out then pay him the half of the value and put the house up for sale and have someone else buy the house. Just becuase he wants to stay if he can not afford the home (based on the bank not approving him) then he can not purchase something he can not afford, end of story. Many people want to live in homes they can not afford and have to settle for something smaller in another location.
If you can not buy him out and he can not buy you out then the home needs to be sold and the proceeds split, him paying you half of the value and him keeping half of the value.