It sounds as if you and your Dad are very keen on this flat and it is doable. I realise your poor Mum won't be able to fully appreciate any reasoning or explanation you might offer but would it make sense for your Dad to describe it as 'our holiday home near our daughter'? Whether or not she can be persuaded to take some (extendable) 'holidays' there he certainly can.
Although a forced change in environment or other circumstances might accelerate her illness I think you could reasonably take a broad view of her best interests and completely losing the care of you and your Dad due to exhaustion or worse is certainly not in her interests.
I'm not sure that you are creating a crisis because you say that both you and your Dad have got to the point of being unable to sustain the current situation. This is just as much a crisis as if your Mum fell and broke a hip. (My partner's Mum with vascular dementia was hospitalised for an unrelated reason, the dementia progressed markedly but there was no choice).
I've come to think that this is not an illness we are battling to halt or reverse, the battle is to create and maintain the best possible family life in the face of it.
I know all our circumstances are different and what I'm saying may be irrelevant.
Changing your focus to put planning your Dad's support needs first and arranging things for your Mum that enable his needs to be met might be a useful shift in perspective for everyone including social services. He is frail and elderly and has the capacity make his own decisions, perhaps it's his turn to come first?
Best of luck with it all,