My mother is in a care home and is self funding. I have finance POA but not health, but that has not so far been a bar to me providing whatever type of care I think is best for her. From what you say, there is a dispute between you and the care home as to whether it is in his best interests to stay or move, which is why social services are now involved.
As you have found, it is pointless asking the PWD (person with dementia) what they want, because you will get a completely different answer depending on who asks, and when. Often the person will give the answer they think the questioner seems to want. I am not sure recording what he said will be helpful, Social Services will want to ask him the question when they meet him, and unfortunately it's pot luck what his answer will be, he won't understand the consequences of his reply. I hope you get the right result for him.