You poor thing, that just sounds terrible. Your poor partner as well, I imagine it would be difficult having to accept that own mother is being this way.
I think you've done the right thing by not replying & blocking her on Facebook. It sounds like she wants you to play into her drama. Hopefully if she's not getting a reaction from you both she may move onto other family instead? Does she have other kids? Have they heard from her?
I hope it doesn't happen but if she did turn up on your doorstep I'd get your partner to tell her to leave. I'd maybe have a talk about what to do if she did turn up so you're both on the same page.
In relation to her making things up to Centrelink/Child Safety. People use this all the time saying they'll lie to get others investigated. It's not that simple. Even if she did make a false claim they will look into it & it would be unsubstantiated. If she keeps making false complaints they will come to know her very well. So please don't let this cause you worry.
It sucks that you've got this person in the family causing so much drama for others. She sounds very selfish. That's terrible shes faking being crook to try & get money. I'd keep a copy of the message even if you can screen shot it. If she does start making false claims to other agencies you can show them this & it would help them see her for who she truly is.
I really hope she gets bored after no reaction & moves on.