You are right but what if you ask her what she needs the money for and you might be able to help out. If child is in an activity, pay that activity directly (send check to place or ask if they will take credit card over phone). If you decide to do this, it will be extra that you are doing but you can prove it by the checks/credit card statements and ask for receipts to be mailed to you.
If you send her a check to her, not going to count. If you send it to state, they will hold on to it.
It does not cost anything here and you do not need a lawyer to ask for an increase. In my state, the state child support
services will even ask for the increase with you. Since you are dealing with two different states and the court system it will take months for her to get an increase so she is probably seeing if you will just agree. If you make more money than 13 years ago, it might make more sense to offer to pay for one of the child's bills. At any point you can stop doing it. You can even split a bill, many people split activities and pay them directly. Depending on how much you are willing to do, it can be more than one bill.
HS kids cost money and school items come up all the time. At the same time, if he is old enough he can get a part time summer job by you if he is with you, or by her to help pay for what he "Wants".