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 4"] [FEN "4k2r/2p1p2p/P1P1P2P/8/8/8/6P1/nn2K2R w Kk - 0 1"]
Clearly, in order to checkmate Black in just four moves, White's pawn must be promoted. However, if White pushes their pawn, then Black will castle and there is no mate! Thus, White must prevent Black from castling The only way to do that is with their rook.
But if White plays 1. Rf1?, then Black will play 1... Nc2+!, also preventing a mate in time. Therefore, White's only option to prevent Black from castling is to castle themselves-1. 0-0!
Here is a shorter mate in 3 version. It shows the idea much more clearly
[Title "me, chesstackexchange.com 11/5/2019, Mate In 3"] [FEN "4k2r/4p1p1/1P2P3/1N6/8/8/7P/n3K2R w Kk - 0 1"]