I'm not fond of the FRO but what happens is, you pay the FRO, not her. There is nothing for her lie about. "He didn't pay me last month!" Ummm, sorry, he doesn't pay you, he pays the FRO.
The FRO means that you have even less direct contact with her than before. Just make sure you understand what is going on and when the payments switch over. Get a file number and call the FRO and make sure you are in the loop. What could happen, you pay her for October, and then find the FRO has started charging you that month too.
If something like that happened, while I would call the FRO and get their word on it, I would more likely just take the ex to small claims court, showing proof she cashed the cheque and showing proof you paid the FRO. Keep detailed bank records, get copies of the cashed cheques from your bank, make sure you write on the cheque "Child Support for Sept."
Once it is in place, make sure you stay on top of updating your court order according your income each year. When you get to the point when you think the CS should end, start the process ahead of time.
Always remember the FRO does not make decisions, they enforce existing court orders. There is no use arguing with them unless there is some technical problem like they charge you twice for one month or something. If you need something changed, you go through the courts.