They want you to register and pay, (lying to them and potential emergency services) by telling them you're staying at one of the non-reservable camps. Online only with card, valid email, intrusive questions etc. There is no way to pay cash.
"No-trace Camping" refers to no official recognition that anyone uses the park beyond the contracted-out campground, therefore ignore it. All they care is that you paid.
Go ahead and bivy, done all the time, but best only well beyond the camps and hordes.