That isn't how it works, in any state in the US. Your license would have been suspended BEFORE the IRS intercept was ever put into place. You were non-compliant as soon as you did not pay the support when it was due.
They also did not inform you that the money would be mailed. They informed you that your refund had been seized for child support debt.
Some of this is incorrect. License suspension and Tax intercept are two separate enforcement remedies. They occur independent of one another.
OP--unless your license was suspended while you were current on your support or because you don't owe it and happen to share a name with someone who does, then your license was not wrongfully suspended.
Federal tax intercept becomes active automatically once a payor is behind more than $25. Case gets flagged, and the following month the arrears are certified and sent to the IRS. It's what we call an "administrative enforce the remedy"--meaning the system automatically does it and a worker doesn't have to activate it AND the workers cannot deactivate it.Even if the payor is complaint with a payment plan.
DLS suspension follows another track: A payor must have arrears greater than X amount (varies by state), and have failed to make any payments in the past 90 days. And once the case is selected for DLS, it still takes another 30-90 days before DLS actually occurs. In this time, the payor will be noticed that their license will be suspended, giving them time to contact the CSED to make payment arrangements. (This is where things get very different.) Each county gets to set policy on what an acceptable good faith payment would be, or if they require all arrears to be paid, or what kind of reduced payment plan would be acceptable. My agency allowed us to set such terms on a case by case basis. I met with my supervisor if I was doing anything unusual--requiring a larger good faith payment, setting the payment plan amount lower than the actual support obligation, etc.
OP's problem: He thinks that because he made a deal with the other parent, he's good to go. What he needed to do to avoid DLS was contact CSED when he received notice about DLS and enter into a payment agreement. And he is absolutely right that his intercepted refund would count for his payment as it wasn't voluntary. THe State is going to go after that money because the OP obviously hasn't been paying support as ordered, and based on OP's post, hasn't been for at least 5 months.
To get the DLS lifted, he needs to contact his CSED worker, and enter into a payment agreement. Be expected to pay anywhere from 1-3 months obligation as a good faith payment to facilitate the reinstatement, and to tell your worker how you plan to get and remain compliant with the order.