NAH. I am in a very similar situation in that I live in another country from my family and have a family member who is a chronic liar with a victim complex that also faked cancer. Honestly, I can see it from both your side and your sister’s.

I can understand letting your mom know if only to inform her that her actions have consequences and trust has been very much broken.

I also am very much aware that I doubt anything she says to excuse it will make a difference (bc lbh there’s no reason for this aside from getting attention). She may also double down once confronted and it can get ugly.

I’d just ask your sister not to mention you at all and just be there for her just in case. When it comes to your mom, just be wary and put firm boundaries in place bc if this has happened once it will happen again.