You may want to skip to my next post. This might be helpful to someone else in the future, but I think your issue is probably with the 2nd Coin Relay.
Since it's two player it works a little differently than the four player version. There's no schematic for me to check, but I imagine it's somewhat like what I was talking about with this Williams Olympic Hockey:
https://pinside.com/pinball/forum/topic/olympic-hockey-not-switcing-back-to-p1
Ball Count Unit, 2nd Coin Relay, or Player Relay involved? Maybe the 2nd Coin Relay is latched and not coming unlatched?
It might also be a misadjusted score motor switch. When the game is in one player mode, it pulses score motor switch 1A and score motor switch 2A, which produces two pulses and makes the ball count unit skip the 2nd player parts of the circuit. If 1A or 2A didn't actually close, the game would go to the 2nd player parts anyway. This could also happen if the 2nd Coin relay was latched, because there is a make/break on it that disables the extra pulse. If it was latched, though, the lights should be changing back and forth from player 1 to player 2 and the game should be indicating "2 can play" from the start. That's what makes me wonder about the score motor switch.
score motor switches ball count.png
The circled red switches are the two pulses that make the game skip the 2nd player. If only one of them is pulsed, because the other switch isn't really making contact, it'd only be advancing one step on the ball count unit, regardless of whether the game was in one or two player mode.
NOTE: This is the schematic for Olympic Hockey from 1972 with step ups on 1A and 2A. The schematic for Lucky Seven from 1977 has the ball count step ups on 2A and 3A instead. There don't appear to be any other 2-player Williams schematics on IPDB in between those years, so I can't tell you for sure which switches are correct for Satin Doll.