^ this - try respite both to give your sister a break AND see how your mum responds
That said, MIL is in a care home, self funding, £6k a month - it will eat up all she and her late DH worked for all their lives, & my DP would have liked her to stay in her own home with carers for longer
Moving into a home is stressful, exhausting and tends to cause more rapid deterioration
Your sister needs to be transparent with you about her benefit situation and you need to research your own councils level of discretion
5 years in a care home (which is not unlikely, women with dementia can live a long time) will cost about 350k - if the house is taken neither of you will get much of anything down the line
Look at respite, day centres, carers etc as other options and fill yourself in on the finances, legal picture , your sister’s entitlement to stay in the house AND your mums will
Its a health/social functioning situation but also a big decision with long term implications for your family’s finances
Could you mix a stay at yours and respite while you look into things so your sister has a break?
Agree with PP that adults with dementia try REALLY hard to cover up and can be quite convincing. A stay with you of eg a month could be quite revealing
Also consider implications of travelling to visit. MIL’s care home is 45 min away, so a visit will be 3 hours or so (half a day gone). DH goes every weekend and 1 evening a week, it’s exhausting after a long days work in winter but no option
if she were home with carers nearby it would have been so much easier to go frequently