Update. I've run with this solution for some time now and have been in some pretty rough water. The rubber bands worked well, if the door popped loose the rubber band would pull it latched again, but my first mate 'forgets things' and would break the rubber band when opening the walk thru doors. Running out of rubber bands, I rigged up some bungee cords:

The bungee cords were a bad idea. Rather than breaking a rubber band, the forgetful first mate broke a clip off. Also when I hit a rather large rogue wave just right, the bunge pulled a clip loose. I'm still happy with the solution, it really helps me keep my focus on driving the boat rather than fussing with doors that popped loose, but I'm going back to rubber bands.