Honestly I would not let her go home on her own unless you absolutely have to. I’ve been through this myself and it was an utter disaster, we should have absolutely insisted our relative went into sheltered accommodation/ a home well before they did.
In theory she may have capacity and in theory yes you could let her do what she wants and set boundaries, but in practice it will not work. With her diagnosis and at her age, if she goes home without care she is likely to deteriorate really quickly and make everyone else’s life miserable to boot. My relative with an almost identical diagnosis quickly became unable to use the kettle/ boiler/ cooker, forgot to eat and drink, lost tons of weight and had UTI after UTI, became incredibly paranoid (locking herself in the house), accused anyone she met of stealing from her. Was unable to deal with anyone on the phone let alone the dreaded internet.
I’d be getting distressed calls from her at all hours, from concerned neighbours and other people and she was thoroughly miserable. We eventually got her into a lovely home (basically the GP had to make an emergency mental health sectioning) but she was so thin and ill by then she didn’t last long.
I think you need to give her a few options- either staying with you and your sister, looking at sheltered housing, or at home but with a suite of carer visits and powers of attorney etc so someone else deals
with her finances. Make it clear that if she insists on going home she will very much be on her own and it would be incredibly selfish to you and the rest of the family and, in all likelihood, she’ll end up in hospital soon and they won’t discharge her to live on her own. Seems mean I know but she’s got to face reality and realise that her wishes don’t trump everything else. It’s like a toddler- sometimes you have to do what’s best for them even if they don’t like it.
Also do make sure you’ve got a power of attorney (medical and financial) whilst she still has capacity and also that her will/ funeral arrangements are settled. This might be the last few months where you can do it (hopefully not).