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I'm pretty sure you can't just have a Drunken Elephant. If there's no Prince, then you've already lost if your King is already mated.

I have made a 3D-printable Chu Shogi set, which you can find here on Thingiverse.

Quick question: what are the specifics of check in this game? Is it when at least one of your royals is under attack, or is it when your only royal is under attack?
The answer has consequences for the repetition rule.
Wikipedia says the latter.

Chu Shogi has no checking rule. You win by exterminating all opponent royals.
What will happen if the opponent captured your King then you can get a Prince from a Drunk Elephant immediately?

What will happen if the opponent captured your King then you can get a Prince from a Drunk Elephant immediately?
You lost. There is no 'aftermove', the game ends at the end of the turn of the player who captured your only royal. Likewise, if you capture the opponent's last royal with a move that exposes your own last royal, he cannot draw by capturing that and leave you both kingless.
That is the essential difference between royalty extinction and baring (as in Shatranj). In the latter case you do get an aftermove to avert the loss, and turn it into a win (by capturing the King) or into a draw (by counterbaring).
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But what if you left only Drunken Elephant and it has been captured before its promotion?