I wouldn't take her back based on what you wrote, but I'm not in the middle of the situation and only have the one side. Why does she keep lying? Do you think she can stop?

The thing about her telling people you smacked her around is really, really bad, worse than cheating, imo. A neighbor of mine had a false report filed on him several years ago and it has complicated his life. He could lose his job if she does it again, as well as access to his kid.

Another friend is going through a situation right now where he called the state because his GF was physically abusing their child and she called the cops on him to retaliate. It's turning into an extremely ugly breakup, which is probably why I am seeing red flags with your GF. I might just be hyper sensitive about this right now.

I hope that it all works out, whatever you decide to do. Perhaps it would be worthwhile to talk to someone about custody rights even if you do take her back.