What is a more difficult problem which involves using the king to prevent castling than the following problem?
White to play and win:
[FEN "r3k3/5p2/2K2P1P/b1N1P2P/8/8/4n3/8 w q - 0 1"]
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In this famous problem by H. Hultberg (1944), the white king castles to prevent Black from castling:
[FEN "r3k2/1p1p2p1/2pP3p/8/8/5R2/PPPP4/4K2R - - - 0 0 "]
White to mate in two moves.
According to chess problem conventions, castling is presumed to be legal unless it's provably illegal. In this position, you can prove that at least one player has lost the right to castle, but you don't know which one. (If the white rook on f3 came from a1, the white king must have moved to let it out. If the rook on f3 is a promoted piece, it must have visited a8, e8, or f8 before escaping the 8th rank, so the black king or rook must have moved.)
1.Rhf1? fails to 1...O-O-O.
1.O-O! and 2.Rf8#, as Black is now unable to castle.
In a comment, the OP asked for a non-retrograde analysis problem in which both castling rights are known, and, for fun, I have done so.
[Title "Me, chesstackexchange.com 8/27/2019, Mate In 3"] [FEN "4k2r/4p1p1/1P2P3/1N6/8/8/7P/n3K2R w Kk - 0 1"]
Clearly, in order to give mate in time, White's pawn must be promoted. However, if White makes a pawn push, Black castles and there is no mate! White must prevent it, and only their rook can do so.
But if White plays 1. Rf1?, Black plays 1... Nc2+!, delaying mate. Therefore, White's only option to prevent Black's castling is to castle themselves-1. 0-0!, and Black will be mated in two moves no matter what.