If its any consolation the worst times may in retrospect be the funniest.
Wynken. Bursting into tears is a sign your resilience is shot. You know that. Plus you are putting your energies into enabling both your mother to remain in an unsustainable position and your brother to pretend nothing is wrong and that he can stay on the other side of the world making decisions for you.
My guess is that any conversation with siblings is subject to all sorts of history. Is your brother older or younger? Was there any rivalry? etc.
We all have to wear two hats, that of a "manager" and that of a relation. The manager (POA) needs to make the right decision, independent of any emotion the relation might be feeling. However your brother may not be able to separate the two, especially if, as so many brothers seem to, he wants to avoid accepting there is a problem.
(Actually I prefer the word Advocate as this better describes the role when dealing with SS, hospitals etc.)
My experience was of several awkward conversation with my brother over the three years when my mother was struggling, where I was looking for support and validation, something he could not give as he could not acknowledge there was a problem or such a problem that he needed to engage. Plus a little frustration on his part that when he did offer "advice" he felt I did not listen, true in that he was not there and so to me his advice was not informed by reality. Plus I suspect my mother was more likely to tell him she was fine, and tell me she had problems.
I would be tempted to set options down in writing. If nothing else it will clarify your own thinking. You might even discuss it with a friendly health professional. You and your husband then send it to him acknowledging that it is slightly formal but both you and your husband feel that the time has come when decisions need to be made. Not making a decision is a decision. Your mother has dementia and cannot be relied on to take everything on board. She is understandably anxious and likely to react to decision made over her head, but this is the time when in her own best interests you may need to pull together and do this.
Option 1. Stay where she is. She needs over night care. Hugely expensive, and issue of whether she will accept. You have done as much as you can but have got to the stage where you are going to have to draw boundaries for the sake of both you and your family. This will involve you unplugging your phone at night time, leaving her reliant on the pendant and emergency services.
Option 2. The cottage. If one is available might need to move quickly. More chance of her then settling, than if her health has deteriorated still further. Plus danger if you wait, you lose it as an option. Company, overnight support. You then are freed up to do her admin, provide emotional support and will not be on call. She will be safe.
Option 3. Care home. Too early. Will be institutionalised. Finance? Finding a good one.
Great if you can then say that you have spoken to health professionals. Recognise it is difficult. Say, if appropriate and as constructively as you can, that if you don't hear from him within a week, you will go for the second option. (He may not want to be responsible for decisions that have the capacity to go wrong.)
My guess is that respite will force the issue. They could well then say your mother is not safe to go home. They may say she is not able to cope with the cottage either which would be a pity.