Thanks for your patience Dave.
In school physics, gravitational energy must involve the movement or potential to move through the field. When a pair form close to a gravity-well they will acquire gravitational energy, but until they form there should be no GE, according to old physics anyway.

I clearly have not the new physics needed to explain how gravity could provide energy at some point in the field, and as you say they are able to form without a nearby black hole anyway.

We are only considering your "last case" pairs.
The escapee will gain GE, resulting in mass loss of the hole.
The plungee will lose GE, but it will all be gained by the hole, increasing its mass.

So my sticking point now is "They use some of their gravitational energy to create these particles"