Hi tscottn
You do not want to use the built-in functionality "1 coin - 1 play, the play starts right away", hmm.
Look at the JPG - "pink, rosa jumper, question mark" - I do not know if this works - the big question is "when the coin falls through - is the coin-switch closed long enough in time so the Replay-Counter can step-up ?". You may want to try - unsolder the wire (that finally reaches coil on S-Relay) - clip-on a Jumper-Wire at the freed solderlug on the coin-switch - clip-on the other end of the Jumper-Wire at the 15-pos-switch on the replay-counter - then try: Throw-in a coin - does this works ?

It may not work - then You'd have to do some work "my orange stuff in the JPG". The way the coining up works - lets plug the adj-plug to "5 plays per coin" - see by this "my encircled red, blue, brown switches" are bypassed, overridden" --- the motor makes a turn of 120 degrees, "encircled green" switch motor-1A closes five times - all five pulses reach the Replay-Step-Up-Coil.
When You change the adj-plug to "4 plays per coin" then "my marked red" switch motor-2C comes into action - see the motor sequence diagram in the upper right corner of the JPG --- motor-2C switch is open so the second closing of motor-1A is "electrically suppressed". The same way the adj-plug set to "3, 2 plays per coin" does work.
To get what You want: You must mount an added, new Normally-Closed-Switch on motor-position-1D*** - and Your adj-plug no longer works "5 - 4 - 3 - 2 plays per coin" - it now works as "4 - 3 - 2- 1 plays per coin".
See "encircled orange" in the JPG - Your King Pin machine does already have an switch mounted on motor-position-1D*** --- a switch for making Number-Match-Replay --- so it should be possible to mount the added, new Normally-Closed-Switch on motor-position-1D***. Greetings Rolf

0King-Pin-Work-04 (resized).jpg