There is certainly a real possibility that they would never know about the illegally obtained benefits. However, with your former spouse threatening to report you, there is clearly a reason to be concerned.
I would urge you to speak with a criminal defense attorney in your area. Every office has a different approach as to how to handle illegally obtained benefits. An attorney could contact them and inquire as to whether they would be willing to receive reimbursement in exchange for an agreement to not prosecute.
Even if there were to be a prosecution, you may be able to obtain a reasonable result. As a former prosecutor, I am very familiar with this type of case. As my Info Request alluded to, the prosecutor's primary objective is to get reimbursement for the department. If you can afford to get the restitution paid in full, it is very likely you could obtain a probationary sentence.
Please let me know if anything requires clarification.
I hope you found my answer helpful. If so, please click on the green ACCEPT for my answer. This is necessary for me to be paid for my work and so that I can get credit for assisting you. Even if you are a subscription member, you will need to click ACCEPT. Your question will not close, and you will still have the opportunity to follow-up if needed. Leaving a bonus and positive feedback is not required, but doing so is certainly appreciated! Thank you and good luck!
Also, several customers have asked how they may direct a question to me in particular. If you specifically want me to assist you in your legal matter, just put "FOR JOSEPH" in the subject line and I will gladly pick up the question as soon as I am on-line.