Your father is being irrational. he has reached a stage of his life when he has only has 2 options. Either he stays in his appartment until he dies, but to do that he has to accept a live in carer, or he goes into a care home.
Once he is bed-bound, his preference for staying in his appartment without full time care is not possible because it is not safe and the care agency, quite rightly do not want to take responsibility for giving unsafe care and then being held liable if anything goes wrong.
You could discuss this issue with other care agencies to see if they can come up with a package acceptable to your father, but I suspect the answer will be the same; if he is bed-bound he needs live in care.
I can understand how difficult it is for your father, but like it or not all of us have, at times, to make decisions where we choose not what we want, but what is in our best interests. For him, this is one of them.
You need to talk to your father and try and get him to understand that he can make the decisions about his future care, but the choice must be between what is possible and safe for him, and not keep hankering for solutions that are simply not suitable for a man in his condition.