How can I cope with a sibling who has chosen to keep me "out of the loop" as to how Mom is doing, now that Mom is living with her?

Can't you go and visit your mother or call sister and ask how she is doing? Then you can see first hand what is going on. I am sure your sister is going through the same things with your mother as you did. You want this situation to work so you don't have to be the one stuck caring for her...... Maybe offer to give her a 4 hour break sometime. This may open up the door for more communication.
I know with my sister in law, we had tried to keep a communication book so all caregivers would know what was going on. My SIL rips pages out of the book and constantly sabotages my MIL's care. She is a nut job! We heard from the live in her daughters even told her not to do that. We gave up trying to keep her in the loop. I am sure that is not the case with you as you sound very caring. Enjoy your respite..!

Unfortunately what you are describing sounds a lot like what happened with my mother several years ago. She was so extremely nasty to me that I broke off communication with her for 6 months in order to heal. Many of her friends dropped her as well for her nasty behavior. I noticed that mom told me her 80 year old friend was going to bars and picking up men (not likely) that was mom's excuse for dropping her. Mom said hateful things to other family members as well. Flash forward 7 years and mom kept hiding her purse every time she visited me. Later on she would hide her purse at home and dad would have me drive 125 miles to find her purse. Flash forward 10 years and my mother is now in the final phase and dying of dementia/Alzheimers. Now I know it was the disease talking. I'm afraid this may be what is going on in your family. The earlier stages are hard to detect. You get blamed for stuff you haven't done, in fact, never did. I remember my mom claiming I was trying to buy a home in a retirement community - you had to be at least 55 to move in there and I was only 42 and my husband 44. It was like she wouldn't believe me about this, even when I showed her the requirements online. Also, my husband and I have a very nice house and no real reason to move. It is stressful to deal with and hard to understand, especially if you have never had any experience with dementia/Alz patients.

I pray this is not the situation in your family, but it sounds so familiar, that I thought I should share this with you - only a medical professional could tell you and your sister if this is what is happening and even so, diagnosis is not that easy in the earlier stages. Please read up on Alzheimer's and dementia, just in case as it will help you understand the behaviors and why this is happening. You will all NEED each other if that is what is going on. Do what you can to keep the lines of communication open.

Be there for sister when it starts happening to her, because it probably will, after a short "honeymoon" phase. She probably believes a lot of stories that are not true right now, and there may or may not be a way to accelerate her realizing that Mom is not altogether accurate. Maybe write her a letter that makes it clear that you are grateful for the respite and watch out for______ just so she knows the line of communication could be back open when she is ready.