With extinction royalty you could ignore any check as long as you still have multiple royals. With an additional rule that a King must get out of check when he can, this makes it indistinguishable from the case where the King is the only royal, as the Prince can be kept in check because it is an extinction royal and you still have a King. So I am not sure what exactly the change is you have in mind.
If you consider changing the rule, how about this: the King is the only royal piece, but in the case of checkmate it must swap places with the Prince, (like in Tamelane II succession), in addition to doing another move. The King would still stay the only royal, but it is now where the Prince used to be. If the opponent decides to capture to the old location of the King, this leads exactly to the situation your original rule would have led to if he had captured the King. Only if he doesn't capture immediately Prince and King would have been swapped.
This would be easier to support in the Diagram, where it would just be a small modification on the way it now treats Tamerlane II succession. The code there is only executed in stalemate positions, and these are so rare that it doesn't have any effect on performance of the AI. (So it is affordable to scan the board for finding the location of King and Prince(s).) Currently it then performs the King-Prince swap(s), and searches on from there with the opponent on move, to judge whether the swap (that took a turn) brings any solace. In the proposed case it would have to search on with the same player on move, (as the swap was an automatic consequence of the checkmate), to find the best move to combine with the swap.
With extinction royalty you could ignore any check as long as you still have multiple royals. With an additional rule that a King must get out of check when he can, this makes it indistinguishable from the case where the King is the only royal, as the Prince can be kept in check because it is an extinction royal and you still have a King. So I am not sure what exactly the change is you have in mind.
If you consider changing the rule, how about this: the King is the only royal piece, but in the case of checkmate it must swap places with the Prince, (like in Tamelane II succession), in addition to doing another move. The King would still stay the only royal, but it is now where the Prince used to be. If the opponent decides to capture to the old location of the King, this leads exactly to the situation your original rule would have led to if he had captured the King. Only if he doesn't capture immediately Prince and King would have been swapped.
This would be easier to support in the Diagram, where it would just be a small modification on the way it now treats Tamerlane II succession. The code there is only executed in stalemate positions, and these are so rare that it doesn't have any effect on performance of the AI. (So it is affordable to scan the board for finding the location of King and Prince(s).) Currently it then performs the King-Prince swap(s), and searches on from there with the opponent on move, to judge whether the swap (that took a turn) brings any solace. In the proposed case it would have to search on with the same player on move, (as the swap was an automatic consequence of the checkmate), to find the best move to combine with the swap.