the field of the black hole doesn't change with the test particle. IOWs the space remains unchanged. There is no forward propagating beam that diminishes the field, even to the point of reversing it. No forward beam that reduces the curvature of space.
"Why do the black hole and the approaching payload care what a distant observer sees?"
They don't. There is no repulsive force. The test particle just sees the curvature of space due to the blk hole.
I just realized you were asking rhetorical questions directed at Felber, but I haven't had my coffee yet. :)
If you're traveling in a spaceship at .577c, and you stand on the nose, and drop an object, it is going to fall toward the ship because of the ship's (slight) gravitational attraction. That means it will approach and collide with your ship. Otherwise, the principle of relativity would be contradicted. That principle requires that experiments work the same regardless of the speed at which you are moving.
The "distant observer" will also see the object approach and collide with your ship, although he will measure the time required for the collision to be different. Therefore, he cannot see the object as being repelled.