I would start from the end positions, because this is what matters.

There are 2 tokens and 21 blank squares in your example, which means that in the worst case (all ordered pairs of squares represent reachable outcomes) there are 441 valid outcomes.

Check each of these possible outcomes to look for one valid way of reaching it, once you find one way of moving your tokens there mark the outcome as valid and move to the next possible outcome to check. If you don't find any, mark it as invalid and move on.

At the end of the process you have your list of reachable outcomes and you have avoided computing duplicate ways of reaching the same outcome.

No free lunch though, of course. Now the problem becomes how to search for one way of taking all tokens in each ordered set of end positions. But you have a lot of information to use for making the algorithm smarter than "Check all possible moves and turns and verify whether they end up there". For instance, no need to turn left if the required end position is on the right; or, in your example, the green end position cannot be reached in any other way than coming from the left.

It's going to be tough but it's feasible IMHO. And a fun algorithm to code.

I would start from the end positions, because this is what matters.

There are 2 tokens and 21 blank squares in your example, which means that in the worst case (all ordered pairs of squares represent reachable outcomes) there are 420 valid outcomes.

Check each of these possible outcomes to look for one valid way of reaching it, once you find one way of moving your tokens there mark the outcome as valid and move to the next possible outcome to check. If you don't find any, mark it as invalid and move on.

At the end of the process you have your list of reachable outcomes and you have avoided computing duplicate ways of reaching the same outcome.

No free lunch though, of course. Now the problem becomes how to search for one way of taking all tokens in each ordered set of end positions. But you have a lot of information to use for making the algorithm smarter than "Check all possible moves and turns and verify whether they end up there". For instance, no need to turn left if the required end position is on the right; or, in your example, the green end position cannot be reached in any other way than coming from the left.

It's going to be tough but it's feasible IMHO. And a fun algorithm to code.