How many solvers are there? If there's a finite number of them, then I have a simple solution.

@Laska Since pretty much everything is "critical", I thought it better to include a link for viewing the solution on Lichess. Added a pic though. :-) The king cannot take the h-pawn with knight h4 in place, it would have to walk through a check in order to not be late. Also, moving the knight to h4 after the king has passed through isn't viable; the knight would have to stand and wait in a spot that blocks a pawn move.

Having spent quite some time watching CtC, I think their stance on "reasoning from uniqueness of the solution" is 1: if you are speed solving in a competition, just go for it. 2: if you are explicitly given the information that the solution is unique (like here), you can use it. 3: otherwise, it isn't a valid deduction; nothing in the basic sudoku rules says that the solution to a given puzzle must be unique.