AI strategies for losing positions
Let's assume all possible paths are finite to make the discussion simpler. This is the case for the vast majority of game actually played such as Chess (because of the 50-move rule) or Go (assuming superko is forbidden).

The idea you have about chosing the move maximizing the minimum path is actually an instance of the following more general idea. A proof tree for player P is a subtree of game tree starting at the current position such that for every node n of the tree


if the position corresponding to n is terminal then P has won on that position
if P is to play in the position corresponding to n, then n has exactly one child
if the opponent of P is to play, then every opponent's move leads to a child of n
A proof tree corresponds to a winning strategy for P, so if a position is a theoretical win, then it should be possible to find a proof tree.


Now, there may be several proof trees to choose from in a given won position. To decide which one should be chosen, you can define a cost for each proof tree. For instance, the fastest win is obtained by finding a proof tree of minimal depth. The simplest win is obtained by finding a proof tree of minimal size etc.
After you have decided on the definition of cost, you get a strategy for the losing player: play the move leading to a position with maximal cost. 

Now, the only things that remains is finding the minimum cost for won positions and finding the maximum cost for lost positions. As you have discovered, this can be done by retrograd analysis when the cost is the depth of a tree. It turns out it's not too hard to generalize this for other cost measures still using retrograd analysis.

If your game is too large to perform retrograd analysis or you are only interested in a specific starting position, the problem becomes a little bit harder but you can to use a kind of generalization of the A* algorithm. Shameless plug: I have a paper on this very topic.