Interesting puzzle. I wish I understood more and could help. Just thinking out loud here, maybe someone can confirm if I am reading correctly.
When the ball count unit reaches 6, 2 things happen. It trips the game over relay, killing all power to the playfield coils, and it activates the match feature which is controlled via the scoring motor (to cycle through each of the players)
So the challenge appears to be that you need to trigger it before the game over relay trips. One way to accomplish this would be to add a second (expected closed) switch to the hole. Rewire from the 6th position to go through that switch before going to the game over relay.
This will prevent game over if there is a ball in the hole. Then you can wire the hole eject relay to the position 6 as well, and it will eject, then end game.