The way I understand your checkmate rules is that normally you would have to play like the King is the only royal piece. But positions where you would then be checkmated you have to recalculate after making the Prince (if there is one) a 'temporary royal' instead, i.e. upto and including the opponent's next move. After which royalty returns to your King if it is still there, or the Prince morphs into a King, making its royalty permanent, if it was not. If you also have no legal moves when the Prince is royal, you have lost immediately.
I guess this means that when your King runs into checkmate, and your Prince then delivers a check to the opponent King through his Knight move, the opponent can resolve that check by capturing your King (demoting the Prince to a King, which loses it the checking move).
The way I understand your checkmate rules is that normally you would have to play like the King is the only royal piece. But positions where you would then be checkmated you have to recalculate after making the Prince (if there is one) a 'temporary royal' instead, i.e. upto and including the opponent's next move. After which royalty returns to your King if it is still there, or the Prince morphs into a King, making its royalty permanent, if it was not. If you also have no legal moves when the Prince is royal, you have lost immediately.
I guess this means that when your King runs into checkmate, and your Prince then delivers a check to the opponent King through his Knight move, the opponent can resolve that check by capturing your King (demoting the Prince to a King, which loses it the checking move).