I'm sorry that the carer said that - I think they are wrong. It sounds as though your father is deteriorating generally. Where he is will make little difference to that.
In the later stages of dementia there definitely seems to be a 'giving up' factor. I think many of us have seen it in our loved ones but I really feel it's part of the dementia and not something that can be rectified. (Though some people do benefit from anti-depressants.) As the brain gradually shuts down the person is just no longer able to function as they once did. How could they? There comes a point where the person does seem to give up and goes silent, effectively turning their life over to their carers. Awareness is greatly diminished if not gone altogether.
I've just re-read what I've written and realise it sounds very negative, depressing and hopeless. I'm really sorry about that. But I honestly think you should leave him where he is. I just can't see that the new annexe/bringing him home will change anything. Sorry.