My sister is talking about not taking care of my Mother through the week anymore after 12 years. What should I do?

She is in burnout. We have a big family, but for 13+ years my sister, myself, and her daughter have been the only ones doing the hands on care of our 84 year old mother, after our father died 13 years ago. She is now bedridden, has early dementia, severe degenerative osteoarthritis, wears a brace from childhood polio, and has diabetes. We clean her, bathe her, dress her, fix her meals, fix up her pills weekly, and make sure we use the ointments and other things a nurse or aid would do to take care of her. We basically have to move her from side to side to do the basic care needs and use an electric hoyer lift to get her from the bed into her wheelchair when she has to go to doctor visits. She is bedbound because of a hospital stay 3 years ago where they found a brain tumor (non-malignant) and had to give her high doses of Keppra, which damaged the muscles in her legs. She was walking on a walker before that hospital admission, but can't stand or walk now permanently. My sister has been threatening me that she is going to stop caring for mom and leave at different times because she wants more money and is basically physically unable to do it anymore (burnout). I am moms durable and medical POA, so my sister is taking out her frustrations on my these past two years when she gets overwhelmed. My sister is not welll herself being a pancreas/kidney transplant patient of 4 years so I am worried she will get herself sick if she keeps going like she is. I am thinking about moving mom into a nursing home and selling her house to pay for her initial care, until me know she is approved by Medicaid. I don't want to do it, but my sister is putting undeserved stress on me when she has these outbursts and my health is starting to suffer for it, because I basically have been taking care of all of moms affairs, paying her bills, dealing with doctors and hospitals, and taking care of anything that breaks down in the house for these past 13 years since my dad died. I am at my wits end and need to do the right thing to make sure mom's care continues just in case my sister decides to just up and quit -- leaving my mom with nobody to take care of her during the week. Nobody else in our huge family will step up to help us, so I feel the best thing is to get mom in a nursing home. I am 60 years old now and still work, so I am unable to care for her fulltime during the week. Now -- is this the right thing to do for my mom seeing that my sister is really giving me a hard time and has been threatening that she will leave mom if I don't comply with her wishes (she knows I don't have anyone else in the family who will do it for me).