Welcome to TP, Prezzy, though I'm sad that your family circumstances brought you here. I think your dad should move to another bedroom as you suggest. It will be less distressing for him, and your mum won't have to share a bed with a 'stranger'.
What to say to his wife? It's truly heartbreaking for him. There is some comfort in how much she cares about her husband, even though she doesn't recognise this person as the younger man she imagines her husband to be. Perhaps he could tell her that her husband is his best friend, they are very close, her husband wants her to be well looked after until she sees him again; that sort of thing?
With regard to the bedroom dilemma.... I knew someone who had both parents with dementia. They moved into a nursing home together. Her dad wanted his wife with him all the time and would get jealous and angry if she was taken to the day room without him. She, however, didn't remember him at all and was very frightened about having to share a bedroom with a 'strange man'. She wouldn't allow him to see her in her nightie, so providing personal care for 2 physically frail people in the same room was virtually impossible. Her mum became a screamer because by that time she had lost her speech and it was the only way she could communicate.
The wife was moved to another floor to which he did not have access, but she would be brought up in a wheelchair to visit him at his insistence. Eventually this was stopped because she found it too distressing. He would 'resume control' of the relationship and behave to her as her husband by expecting kisses and hugs and talk about the family. She, poor thing, had no idea who any of her family was and just wanted to be left to get on with her new life without having to see the old man every day.
I know your parents are a long way from that scenario, and I don't want to upset you further. My point is that her dad wanted to keep his wife grounded in the marriage by daily contact, but the more he did this the more she wanted to get away from him. It was the dementia that did it. They had been a close and loving couple when they were both well and he had been her carer for many years before he himself became ill.
I would say that your mum needs more privacy, especially in the refuge that is her bedroom. Our bedrooms are private spaces, where we feel safe and can relax and go to sleep. Would a move to another room by your dad compromise your mum's safety or toilet needs?