I got some useful comments from "MIT Marine" and others in response to Battle not over yet, below. He says to pick your battles. He agreed to counseling, but not to admitted false charges of domestic violence.
In my experience, it never pays to admit to a false accusation in family court. I once admitted to having some unorthodox parenting approaches, thinking that it was a harmless admission. Dr. Gay used it as an excuse to reduce my custody, and to send me to a re-education camp. The family court repeatedly accused me of being "unorthodox", as if that meant something.
Since then, I took a simpler approach. I say that I am always willing to improve my parenting practices, if some expert can show me how I can do anything better than what I have done. So far, none of the court experts could say how I could do anything better than what I did.
For example, I had a visitation supervisor who was supposed to be a therapeutic supervisor. She would write some really kooky criticisms in her reports. She always tried to have at least one criticism in each report. One time after a 4-hour visit, her criticism was that I wanted to play the Monopoly board game for 2 hours and do something else for 2 hours. She insisted on playing Monopoly for 4 hours. I told her that if she is going to criticize me for this, then she ought to actually explain to me why it is better to play Monopoly for 4 hours. I tried to ask her, but she was unable or unwilling to explain it.
As my reader said, I have to pick my battles. There was no use arguing with a high-priced babysitter about the merits of playing Monopoly. We played for 4 hours. Maybe I should have just admitted that I was wrong to want to only play Monopoly for 2 hours. Then maybe the supervisor would have said that I learned something. I don't know what I would have learned, but she probably would have preferred that.