If you are paying CS, I presume it's also under a court order. And if you are paying CS under a court order, then somewhere, you most likely have a copy of the visitation order as well.
Of course, if you are talking about paying child support and getting visitation based on a personal agreement with your X, that means nearly nothing at all.
So, do you have a court order, or not? If so, do what it says, don't do what it doesn't say, and forget about the cage rattling exercise. If you don't have a court order, then you should consider getting one, especially if your X is playing "terrorist" games.
Reminder: the United States Government has a long standing policy of not negotiating with terrorists. Neither should you.
Good Luck. Stay in touch, because your situation isn't about to end anytime soon. At least your 16 year old should be able to bend the judge's ear a bit, if you can push the case there.