I would say get paid carers in, but it sounds as if your Dad is fairly functional if he can still dress and so on. My mum has paid carers 4 x day and it costs about £1,800 a month.
But carers visiting for half hour or 45 min visits really only give medicine, help with dressing, serve meals. Your Mum would still be tied to the house most of the time.
If they have a decent spare room and bathroom in the new house, maybe consider a full-time live-in carer. This is more expensive but probably better value in that it would give your Mum more of her normal life back.
If your Mum won't spend on this, make it clear you can't fill the gap.
You can visit your local GP surgery and ask for leaflets on dementia services in tbe area. We had a day centre about 8 miles away which Mum was booked into for about 5 hours a day once or twice a week. Cost was about £30 per day plus transport. There were activities, talks, lunch, visits by musicians and so on. Best to start these things early on, as once doubly incontinent, people usually can't attend any more (too much work for staff/volunteers). Sometimes local care homes offer this too. Really hard to find out about until you start asking around. Adult, Social Care, Libraries and even Hospital Discharge team may have lists of local services.
There are other clubs/dementia cafes your Mum could go along to with your Dad, where he would do activities/music/exercise and your Mum sit with other carers for coffee and chat.
Or perhaps a volunteer visitor scheme to come to the house for an hour a week and chat to your Dad or.do hobbies with him. A local church does this, but there's also a secular scheme.
I think there are dementia choirs, art groups, football, cricket, bowls etc as well. If your Mum doesn't want to take him,there might be a community transport scheme.
Building in a couole of these a week might help your Mum and gradually encourage her to see that Intervention is helpful and that different things will needed as your Dad's condition progresses.