Hello, new here… serious advice needed

So my sister is a textbook narcissist. I don’t know when we (the family) realized it. But at some point it became abundantly clear.

Not to get into too many details, I’ll get into this specific situation I need advice on.

She was living with my mom, after a breakup, with her 4 children. She found a new man, and moved in with him. On the way out she got into a fight with my mom’s husband, our stepdad of two decades.

She blamed my mother for the fight. She kept her kids from my mother and the entire family for over a year while this guy and her lived together.

Now, finally, they broke up. As I knew would happen, she immediately reached out to my mom as if nothing happened and started asking her to babysit. Because she needed someone, all that didn’t matter. No apologies, no nothing.

Now she’s involved in DCFS etc but that’s another story.

For example tonight, she’s asking her to watch them while she goes and hangs out with some dude.

My mom thinks that just giving in and doing whatever she wants is best because it means she gets to see the kids.

I hate the idea of her not standing her ground though I understand her feelings.

Is there a way to genuinely get some upper ground here on my mother’s end? Should she take a stand and try to force some humility now, while she has something my narcissist sister needs? (Childcare)

Or should she just keep bending over backwards at the drop of a dime every time she calls needing a sitter?

Please help. I’m at my wits end here.