The black hole has more energy inside it than the component parts. That energy includes the gravitational binding energy, which is not "inside" of the black hol

The black hole has more energy inside it than the component parts. That energy includes the gravitational binding energy, which is not "inside" of the black hole. It never passed through the event horizon to get there. So when the black holes merge, they lose some of their gravitational binding energy as the insides fall in to one another. Nothing that passed through the event horizon gets lost. Another way to look at it: imagine you have two counter-spinning black holes. When they collide, the net momentum is now zero, but energy must be conserved. That angular momentum was never really inside the black hole in the first place, so nothing passes through the event horizon to escape it. The gravitational binding energy and angular momentum are different kinds of energy, but they illustrate the same idea.