From what I have seen, the only chance a father generally has is to be the first person to make an accusation (to be the plaintiff). Otherwise, he will be dragged down with endless, baseless, allegations which he must prove were lies with a lot of documentation. Then when he proves she lied (and nothing will possibly happen to her for committing perjury), she will make another accusation, and it repeats. He can attempt to enter motions, but good luck having them heard. The day of the hearing, she need only come in with another accusation of being in fear, and the process resets with his motion set aside to deal with the (obviously) pressing accusation.
If he can get the kids out of that equation immediately, and immediately extricate himself from her vicinity, then her ability to make accusations will be greatly diminished. But also would have to establish right up front her lying and cheating in order to paint a picture of a dishonest and manipulative person.
I am honestly surprised that so many men are still willing to marry American women until this situation is rectified. If you are a professional and even modestly successful man, it makes absolutely no economic or social sense for you to marry a woman in this country. The risk is far too high. Family values issues aside, we are better off simply becoming single fathers with some kind of surrogate program. The only things that make that unrealistic are family values which require a mother and a father. It’s basically the reverse of what women faced in the 19th century.
That’s the state of family courts in this country and why I think the only chance a father could possibly have is to be the plaintiff. Even then he has to have overwhelming evidence that she ought not be given full custody of the children.