This is part of why I'm wanting to also implement the King and Prince as both royal, at least for purposes of the extinction royalty rule.
I don't quite know the rule for declaring check or checkmate with extinction royalty, but my take on it is that it isn't required (except on the last one) but is good to do as a courtesy.
In any event, if the White King is in check, then the White player is strongly encouraged to get it out, though if he has a better move (such as capturing the Black Prince, or otherwise doing serious damage to Black) then he may take that. Likewise, Black is not required to take the White King if he has an opportunity, though he may also find something better to do (like capture or trap the White Prince, capture the White Queen, or just leave the White King where it is because if the White Prince promotes to King it'll be in position to move and capture something in an orthogonally neighboring space that Black doesn't want captured). Only when the Prince is off the board, either by being captured or by promoting to King when the King is captured, do normal check/checkmate rules apply for that side.
That means that, if White still has his Prince and Black doesn't have his, White can deal with his King being in check by putting Black's King in check. Black is then obligated to get his King out of check.
And, as I mentioned, the promotion of Prince to King only (and immediately) happens when the opponent actually makes the move of capturing the King.
So, basically, your scenario is correct but for a couple of minor details, the main one being those generated by the extinction royalty rule.
This is part of why I'm wanting to also implement the King and Prince as both royal, at least for purposes of the extinction royalty rule.
I don't quite know the rule for declaring check or checkmate with extinction royalty, but my take on it is that it isn't required (except on the last one) but is good to do as a courtesy.
In any event, if the White King is in check, then the White player is strongly encouraged to get it out, though if he has a better move (such as capturing the Black Prince, or otherwise doing serious damage to Black) then he may take that. Likewise, Black is not required to take the White King if he has an opportunity, though he may also find something better to do (like capture or trap the White Prince, capture the White Queen, or just leave the White King where it is because if the White Prince promotes to King it'll be in position to move and capture something in an orthogonally neighboring space that Black doesn't want captured). Only when the Prince is off the board, either by being captured or by promoting to King when the King is captured, do normal check/checkmate rules apply for that side.
That means that, if White still has his Prince and Black doesn't have his, White can deal with his King being in check by putting Black's King in check. Black is then obligated to get his King out of check.
And, as I mentioned, the promotion of Prince to King only (and immediately) happens when the opponent actually makes the move of capturing the King.
So, basically, your scenario is correct but for a couple of minor details, the main one being those generated by the extinction royalty rule.