HeyDuddy I am really sorry to hear what you are going through 
I had a similar although less extreme situation with a family member.
My aunty was dying and in a nursing home. My dad and his sister were NOK but very elderly so I was dealing with the home and doctors with their blessing. Aunty had a DNR and, in her lucid moments, extremely vocal about not returning to hospital and wanting to die "in her own room".
One of her friends was horrible to my parents - accusing them of wanting Aunty out of the way and how they had "always hated her". It went on for weeks and was increasingly extreme. It was truly bizarre.
I eventually blocked her from my parent's phones and what's app and left strict instructions with the nursing home that this person was not to try and contact Aunty. Apparently she had been coming to the home and trying to gain entry to bring her food and flowers (it was during lockdown and nobody was allowed in).
There were some mutual friends but fortunately they were all firmly on my parent's "side".
I didn't tell her when Aunty died and I didn't invite her to the funeral. Numbers allowed at funerals were seriously limited due to Covid anyway.
It was horrible and unnecessary behaviour at a very difficult time for my family. I'm still quite angry with her and if I met her in passing I would be giving her a piece of my mind.
I would honestly block this friend of your MIL OP. Do not tell her anything because she won't believe you anyway and you have way more important things happening. Don't feel guilty about doing what your MIL wanted you to do.