If this is to work in any way, the lying has to stop, and right now. As you haven't stated why things are dangerous for her at home, there might be some very good reasons about why she is lying to her BF (like if he was being abusive towards her).
I think that if you want to help her you need to help her get out of that dangerous situation - support her efforts to get out with her kids and find some place to live where she can be safe. Maybe she could use some counseling, too.
Moving in with you and stepping up the poly relationship might not be the best thing, but then again, if it gets her out of a bad h9ome environment it might help. (But then again it might just move the drama to your household, if the BF starts to take it out on you).
It's probably best that you don't share the specifics here, but it's difficult to give concrete advice without them, you know?
This is a very tough situation for you, since you both obviously care for her and want what is best for her and her children.
"Listen, or your tongue will make you deaf." - Native American Proverb