Well, at least it's consistent. It's only happening when scoring, so it is unlikely connected to the search relays.
Note that this problem will be a bit tricky to track down, so patience will be required. (Not that you haven't been patient thus far! Just note that this problem is not a quick fix.)
I would suggest trying the following first:
Examine the back of the selector lock relay - make sure there are no switch tabs touching, broken wires, etc. This is on the trip bank on the back door and is normally labeled. There are two stacks of switches and each switch must operate properly. If you have, for example, a single-pole double throw switch in there that connects the scoring circuit, and also disconnects it, that could cause the short. Your manual will have an explanation of every switch on the selector lock relay and what it does. Look at this and see if anything matches up kinda sorta with the behavior you're seeing and start there.
Examine the search index relay. This has a sand wound resistor that can sometimes fail open or come unsoldered. Normally, this doesn't send it to tilt, so check that area carefully. Be very careful not to poke those switches, they are quite delicate.
We basically have to isolate each 50V coil in the scoring system - which is all of the scoring coils Replay counter step ups, search index, search relays, and selector lock.
The selector lock is something I highly suspect as it happens right when the ball leaves the gate with a winner.