I think your plan is a great one. But lacking in good TV drama.
Wrote the first in a rush.
I would make drama.
I would have cops all over the place...
But in the woods about three miles away. TV cmeras showing how they are beating the brush.
Some woman on TV all excited telling how she saw this guy pulling a kid behind him behind her house about 3:00 AM (her dog’s barking woke her). Last seen going into the woods behind her house.
I would have everyone convinced the guy was on the loose.
And the real BG would be convinced he is the world’s smartest guy.
The only rub to the idea is how to keep the kid medicated, if his medical problem is that serious.
Keeping him medicated would require that the decision on how to proceed would have to be made in the very first hour or two so that the first medical delivery would be large enough that refills would not be necessary.
Unless the negotiator can convince this guy that he needs to come out and change the world with his public appearance, this will likely not end well.