OP this sounds awful for you and your DH, and obviously for your SIL too.
I'm not going to comment on her because we can't know what's making her behave like this or believe these things. I don't want to guess, but I agree it sounds like she needs professional help and diagnosis.
I just wanted to say I understand a little of how you feel because my PILs have done similar things at times, although I don't think they have done them for the same reasons as your SIL.
They once reported DH missing because he hadn't phoned them for a couple of days. He thinks they did it to punish him for not calling, rather than because of any genuine concern for his wellbeing. He is in the armed forces and they reported him missing to the ward room where he was based. He was a fairly new recruit and it got him into a lot of trouble, including being confined to the base for several days. And, it being the armed forces, he also had to put up with a lot of teasing about "telling his mummy" where he was.
Not long after we were married he needed a small medical procedure as he had an infected cyst on his shoulder. This meant I had a full week of hysterical phone calls accusing me of lying to them because they knew he really had cancer and we were keeping it a secret, and they were never going to forgive me for not telling them the truth.
More recently (after a very long story and many, many more things like this) we have stopped all contact with them. DH has made his own decision about this but somehow it's become my fault. Even though he works a couple of hundred miles away from home and only comes home at weekends, somehow in their world I am still forbidding him from speaking to them, monitoring his phone, blocking their calls, and reading his messages. Even when he's not here.
They sent him a message reassuring him that they would keep all contact between him and them a secret from now on, and suggesting he deleted all messages and his call log, so I couldn't read them and cause any trouble for him.
He told me about this, just in case you're wondering how I know.
I know it's not true, DH knows its not true, but they are telling anyone who will listen how I'm keeping him from them and forbidding him to call them, and causing trouble for him if he does. It does bother me, and I do find it stressful and upsetting.
So I do really feel for you and your situation with your SIL. Whatever is causing her to behave like this, it doesn't make it any easier for you.
You've come into the family relatively recently, there are people closer to her who might have been better placed to help her long before now.
You can only go by what her family have done and said in the past, and how your DH feels. As her parents have seemingly tried and failed in the past, her siblings are refusing to acknowledge any need for help, and her DH can't seem to see that she needs it, I really don't know what you can do.
Especially as SIL herself is unwilling to get help and posters with experience in similar matters are saying it's almost impossible to get help unless she asks for it or causes real harm.
Of all the people who could have helped her in the past, or who might help her now, you're not the one best placed to do it.
It's hard to live under constant pressure and demands from someone else, even if they can't help it, especially when they seem to genuinely believe the lies and stories they are telling about you.
I think all you can do is respect what you DH decides to do, and if he's serious about cutting contact then maybe changing your phone numbers so you aren't receiving a barrage of accusations by text will be the next step.
Ideally though, your DH should let his sister and her husband know that you are doing this, and why, and hopefully her DH will do something about getting her the help she might need.
I hope you and your DH are okay 