Per wikipedia, half of all 15 puzzle configurations are impossible to solve: https://en.wikipedia.org/wiki/15_puzzle
According to the intelligent people over at Puzzling StackExchange, the puzzle is solvable iff the number of tile swaps (picking up and putting anywhere on the board) is even: https://puzzling.stackexchange.com/questions/25563/do-i-have-an-unsolvable-15-puzzle
Looking for a random-seeming impossible configuration, I stumbled upon oneslime.net's online game: https://oneslime.net/15/
By my calculation, it takes 13 swaps to solve, so is impossible. I remixed the blank version of DavidPhillipOster's “Print in place, Parameterizable, 15-puzzle” here and used the text deboss tool in Prusaslicer 2.6.0 alpha v4 to add the numbers from oneslime.net.
Hope you enjoy! I'm using this as a puzzle for a D&D session, we'll see how long it takes them to figure out…
The author remixed this model.