You absolutely should not move her privately and make yourself responsible for her fees. Your home LA might well refuse to pick it up if you did that. Plus you wouldn’t seem to really have any legal basis to decide to move her without having LPA. There has to be a BI meeting to agree that as she lacks capacity (it likely is in her best interests)
Why the mention of CHC? Is she eligible for CHC? Very few people are. That’s a whole different NHS pot of money and team to deal with. If the LA are already funding it’s no advantage to her to get CHC so not worth bothering from your POV.
I can’t really understand why they have said you should take her out privately and I think it’s incorrect advice and that you should ask to speak with their manager or make a formal complaint.
It’s not at all unheard of for people in care to move out of area nearer their family. The LA currently paying has to continue and, like I said, they are usually pleased with a move out of London as it saves them money.
I can understand them not seeing this as an urgent issue as she is already somewhere safe and cared for from their POV whilst they have lots of folk at home/ in hospital who are not but I don’t think they can say it’s not their issue at all. They need to assign a social worker.
DOLs might help you here? You will be her RPR under DOLs and you could say that in that role you object to her being deprived of her liberty in the current care home as not being in her best interests and request a BI meeting be set up