you don't need to run the puzzle backwards to generate solvable positions, you run it forwards and generate the pieces from the solution moves.
Starting with the piece and square from the previous board, mark all squares but this one as unknown. Make a random move. Any unknown squares passed through get marked blank in the current board and in the puzzle. Randomly choose to keep the same piece (making this square blank in the puzzle) or switch (marking the chosen piece in the puzzle, but now blank in the current board). Continue making random moves (biasing slightly towards choosing unknown squares), eventually stop and randomly assign the remaining unknowns as obstactles or blanks. Because you've said you don't care if there's only one solution, this gives you solvable puzzles trivially.
This isn't a new idea, eg https://www.snellman.net/blog/archive/2019-05-14-procedural-... talks about generating solvable puzzles using the forward moves, you'd generally pair it with a filter that weeds out puzzles that are not fun for some reason (eg, solutions that random walk blank squares; these can be avoided at generation time by having a budget for move-to-blank)
Starting with the piece and square from the previous board, mark all squares but this one as unknown. Make a random move. Any unknown squares passed through get marked blank in the current board and in the puzzle. Randomly choose to keep the same piece (making this square blank in the puzzle) or switch (marking the chosen piece in the puzzle, but now blank in the current board). Continue making random moves (biasing slightly towards choosing unknown squares), eventually stop and randomly assign the remaining unknowns as obstactles or blanks. Because you've said you don't care if there's only one solution, this gives you solvable puzzles trivially.
This isn't a new idea, eg https://www.snellman.net/blog/archive/2019-05-14-procedural-... talks about generating solvable puzzles using the forward moves, you'd generally pair it with a filter that weeds out puzzles that are not fun for some reason (eg, solutions that random walk blank squares; these can be avoided at generation time by having a budget for move-to-blank)