How enmeshed is the SD with her mother?  Is she able to tell the truth about her her mother is to the authorities or is she spreading rumors at school?  What's your DH think about all of this?  What is he doing or not doing?  I'd keep the girls separate as much as possible right now.  Even if SD isn't involved, she needs to know how serious this whole thing is. LIes have consequences.    Both girls are old enough to understand how serious this is.  

My experience with girls, and I have 4 of my own (3 over 21) and 2 SDs is that mean girls are raised by mean girls.  Its really crappy that you have to put up with this from BM.  I'm sure you don't want your BD going around defending herself by dragging SD's mom thru the mud, (even though BM is a hot mess from the sounds of it.)  

I'd consider consulting with a lawyer in your area and see what can be done and how to protect your daughter.  

I'd also educate yourself on personality disordered people, the smear campaign, and how kids bully online.  

I'm really sorry you have to suffer thru this.  I was on edge while my girls where in high school with OSD.  I just never felt totally comfortable that BM and OSD wouldn't pull something on my girls.  Any rumors were about my OSD and she brought that on herself.  

Your job is to protect and champion your daughter.  DH's job is to parent his daughter how ever he needs to so she doesn't end up like her mother.  (to the best of his ability) 

I would seek legal advice. This is an actionable offense through either civil or criminal courts (or both) depending on your jurisdiction. You need to decide if this is something you want to do or at least threaten to do. I'm a go-along-to-get-along kind of person but you have to draw a line somewhere, and I'd say - this is it. 

You may even be able to seek initial help through a charitable 3rd party child protection service who may be able to refer you to appropriate legal advice. 

Bottom line is - the power of the state was brought down upon you and YOUR child - maliciously. 

Why is your daughter having to fight false rumours? Are these rumours being spread on social media? I mean - these things are meant to be anonymous right? How do YOU know BM made this claim? CPS wouldn't normally tell you - so there must be some other info floating around out there that makes you think this is so. There may also be a claim you can make about cyber-bullying, which again may be a criminal offense depending on your jurisdiction. 

It’s not likely anything will come of one false CPS report - BM would have file multiple false ones for anyone to care. 

But you have a traitor in your home, one who is nice to your face but does stuff behind your backs to please BM. That’s tough to combat. 

Everything said is dead on. 

1. Mean girls are definitely raised by mean girls!

2. You have a traitor in your home.

First, forget BM. She tried. She failed. You have steps in place to ensure your daughters safety. If BM pulls this stunt again then I would go after her full force. 

But secondly you need to confront SD. Did she indeed start this rumor or was this projected onto her by BM? And address is. Either way your DD deserves to live in a safe and secure environment. And if SD is going to continue to threaten that she needs to go.

As for the rumor - these are easily disproved by behavior. Your DD simply needs to present herself in a manner that states she is above it all. Sadly she will quickly find out what true friends are but we all learn that lesson some point in life.  

Good job being on top of your 14 yr old's safety and security. DH should do the same with SD.