I'm in a slightly different situation but have some suggestions that might help. It's not unusual for one partner to want to run away from care needs whatever age it happens. I think it must be a really complex emotional response and a fear of losing what you've had for decades so the insistence that they carry on as normal is a pretty standard response.
My parents never had care, very proud, would never have entertained it and now have someone in every weekday. If there is no one at the weekend they are lonely and the days are very long for them. Aside from the practical support it gives the day a structure and the staff have very much become close to them. They have more independence now not less.
I would suggest that you try to remove yourself from the power dynamic by getting someone else to have the discussion with your dad. Now is probably not the time to address how/why their expectations of you are unreasonable, I would just keep telling him you're there to support him with the services, they're the specialists but offer to help arrange payments and so on. Absolutely agree with PP about getting listed on the bank accounts and seeking PoA, especially if you Mum has moments where she can give consent to this as once she can't you're too late.
It may be best to come from the GP or another service and presented to your dad that the services are support for his needs as s carer, that your mum has been allocated funds to get specific help for her needs that he can't be expected to give. Talk to Carers UK and Age UK, they have tones of experience with this.
How invested are they as grandparents, would they put the grandchildren's needs above their own? If you were to tell them that dgc need you at home in the next few weeks but you've heard of support locally would they be more amenable?
Age UK provide 'cleaners' (£17.50 ph in our region), we had about a 4 week wait. They can clean, do laundry, shop, iron and so on but most importantly they build a relationship with the family and can provide companionship. They have seen everything and will have a better handle on how to approach your parents than you would expect. Apparently, two hours twice per week is average. Hopefully the Attendance Allowance is for both night & day - around £376 pm? - and it is intended for exactly this kind of thing.
We also use Home Instead, a national company providing care in homes. They are really expensive £22-28 ph but only have a 4 hr per week minimum so that can be still be covered by your Mum's AA. They have a sliding scale of care from housework to companionship to personal care and advocacy (making appointments, calling services and so on).
I think as other PP have said that your mum maybe happier in a residential service for dementia patients. I hear it can be worse for patients at home as there are so many things that remind them of their failing memory and ability. There are some really wonderful homes out there.
I really had a moment where I realised I had to stop 'saving' them and start supporting them instead. I'd try not to react to your dad with anything but neutrality and to be sympathetic to him, he'll likely find it easier to accept your support/guidance then. I truly wish you all the luck.