Using a third party to resolve sibling conflict is often a good idea. Geriatric care managers can knowledgeably assess the situation and offer options.
Yet, your friend's fear of going
behind her sister's back to find a neutral third party is reasonable.
First, share with your friend, that when a primary (hands-on caregiver) feels frustrated and overwhelmed, the last thing s/he needs to hear, are suggestions for caregiving from a not-as-involved family member. On the other hand, if your friend's sister is getting angry, your mother's safety becomes a more pressing issue.
There may be different reasons why your friend's sister is feeling frustrated and overwhelmed in her role as caregiver for your mom with moderate dementia. She may feel trapped by feelings of indebtedness to her mom or perceive the need to honor family obligations. She could even be short on cash and need to share part of Mom's retirement benefits to survive. Whatever her reasons, she is struggling.
Although, your friend's suggestions such as in-home or even residential care are worthwhile options, they are better received if she were also spending time with her mother (if this is possible). Her sister will be more accepting of advice that comes from personal experience. An adult day care center can also provide her sister with respite during the day while giving her the opportunity to attend support group. Being able to discuss caregiving issues with others who walk the same road will uplift her.
In a seemingly "Catch 22" situation like this, the best option is for your friend to be hands-on involved and build her sister's confidence that she'll be by her side to help. Also, she can start calling regularly to get updates. She can ask questions and listen. This acts as a means of much-needed support in allowing her sister to vent. Over time, her sister will be more receptive to her ideas.
Your advice to your long-time friend comes from the heart of a caring friend of a concerned sister whose older sister is experiencing periods of difficulty caring for her mom. It will take patience and involvement by your friend to ensure her mom's safety and to preserve her older sister's sanity.