We have already established that a team that has lost 1 king can keep fighting this game. Just allowing the first king to get captured and the second king capture resulting in a win are quite simple rules.
Four-Player Infinite Chess - The Battle of Morgarten!

Thanks for the support Tomato! Here's the board position. I hope to play my move tomorrow. (my brain is turned off for the day).
39.Q(18,12)+...R(17,11)
40.Q(12,10)+...

He posted a board update half hour ago for my game vs junebug, and said this...
"Now at my brother's house. I have an 8-hour drive and then I'll be home."
...And since the game is over with the first checkmate/king capture there red and yellow wins.
That's a legitimate winning condition.
I'm a fan of trying to survive for as long as possible - even if only one king is alive let him keep fighting. But obviously that makes things more complicated, because you need rules for both when there are two surviving kings, and only one.