Maybe the confusion is that normal chess does not have two kings per army, and we're working from a different premise?
My view is that one alive king keeps an entire team alive.
Your view seems to be that one checkmated king takes priority.
I suppose both views are legitimate, but I take the view that one standing king creates a "right" for all pieces on a team to continue to play normally.

HorribleTomato hat geschrieben: 
Current rule (already agreed):
Kings still cannot be left in check, or placed into check
Expanded rule (to remove ambiguity):
When a team has two kings on the board, and one player has a king in checkmate, the player may not move the king, but may move any other piece.
If it was already agreed that kings cannot be left in check, then no piece should be able to move when a king is checkmated. It would be a rule change, not a clarification.
This must be where there is a misunderstanding.
The rules do say no king can be left in check or placed into check.
But it doesn't say anything about other pieces.
If one king is on the board and not in checkmate, then both players on a team have a right to make any legal move with any other piece.