Can a player resign after checkmating their opponent? A friend of mine recently told me that once, he checkmated his opponent and then immediately resigned because he thought his win was unfair. This happened in a FIDE-rated tournament.
Searching the FIDE Laws of Chess, I found (emphasis mine):

1.2 The objective of each player is to place the opponent's king 'under attack' in such a way that the opponent has no legal move. The player who achieves this goal is said to have 'checkmated' the opponent's king and to have won the game. [...] The opponent whose king has been checkmated has lost the game.



5.1 The game is won by the player who has checkmated his opponent's king. This immediately ends the game, provided that the move producing the checkmate position was legal.

Is this possible in any way, for example, move the piece, don't press the clock, and then resign?
No, this is not possible.

for example move the piece, don't press the clock and then resign?

In particular, that loophole is explicitly covered by the rules:

6.2.1 During the game each player, having made his move on the chessboard, shall stop his own clock and start his opponent’s clock (that is to say, he shall press his clock). This “completes” the move. A move is also completed if:
6.2.1.1 the move ends the game (see Articles 5.1.1, 5.2.1, 5.2.2, 9.6.1 and 9.6.2)

and 5.1.1 is 5.1 in the version of the rules you quoted, so checkmate ends the game before the clock is pressed.
I guess both players could pretend the checkmating move wasn't made (or pretend they don't see it's checkmate), but this is considered cheating, and when discovered may result in both players losing the game and even being expelled from the tournament.