What?! Lying is "No, officer, I don't know how fast I was going". Lying is "no, honey, you look great in that dress". This broad attempted to destroy three young men's lives and, by extension, the lives of their families. They would've been bankrupted defending their kids, providing they didn't shun them outright. And you simply call it lying?! Read up on the Duke false rape case & Mike Nifong's shenanigans.
Been there, done that. I had the ex arrested for DV, didn't file a protective order. A week later, as I'm packing the kids to go to my folks after filing for divorce, she called 911 and had me arrested on false charges. Despite the week difference in the two incidents (and the fact that hers was clearly in retaliation for filing for divorce), the prosecutor handled this as a cross claim. Bias is alive and well.
Rather than a wall of text, let me just say that the longer you have the kids (prior to any hearing) the better. Judges are loathe to upset a situation that's working and the greater amount of time you have invested in this end of things the better. (divorced dad w/ custody as well, I suspect we married the same woman)
Without going into details, you've pretty much outlined my sitch to a t. I have the kids, ex is required to pay. She doesn't and, when she gets a contempt charge for non compliance, it's issued without sanction. Bonus - when I do bother to go into the CS office to see if anything can be done, I get the stinkeye from the drones there - likely assuming I'm the deadbeat.