I think they'll let you do it, because there's no problem with "fairness" on this. You legit have one child already registered, and you want to swap them out for another child in your same family in the same age group. Why would they have a problem with that?
The harder thing to do or the thing that raises fairness questions would be if you tried to add your 2nd kid without taking out your 1st kid.
Call them next week (or even late afternoon this week, once the noon chaos dies down) and ask them how you can do that. If that doesn't work, I agree with a PP, just show up with the other kid and explain at the camp itself. But best to do it officially ahead of time.
And yes, add the other kid to the waitlist now.