I think you should invite her for your FIB sake. You don't have to explain why...
It likely will be difficult to have her negative energy around at a special event but it sounds like she is not social/friendly type so will probably stay away. So long as you continue to be polite then she can't won't have anything to use to create more lies. It sounds like she has some personal issues that she needs to work on. Hopefully this experience will shed some light on how disrespectful, shellfish and negative person she is and maybe realize being like this to others who treat her with kindness and respect don't deserve this.
As far as it stands, she is not going away and future family functions can become very very stressful if she and her husband don't come to the wedding...
what goes around comes around.