Fake path is what I was thinking. Just a quick idea, I haven't tried this, but if your layouts are generally going to be similar to your example it should work...
You're going to spawn four invisible helper objects at each wall object , offset at each corner. Whenever you select a destination, your player will pick the nearest helper object that exists between player and destination, and move there. When it arrives, repeat for the next closest helper object between player and destination (the ones that have been passed are no longer considered). Continue until no more helper sprites exist between player and destination. This should get you a very rudimentary pathfinding system.
There are many situations it might not be ideal though...