That's why I'm not clear on the goal, is it:

Find an algorithm, whose performance is inconsequential, that can determine the minimal number of moves to sort the tapes.

Or is it something like:

Find an optimal algorithm for sorting the tapes such that the only atomic operations move(m,n) and examine(n) -- which tells you what tape is in slot n.

If any amount of preprocessing and analysis is allowed, then any number of hueristics could be useful for find a path from the starting order to sorted order.

If nothing else, you can do a breadth first search of all the possible permutations of moves untill you achieve the desired ordering.