You could try the
Distribute Layers script, though I'm not sure how much control you have in where the layers are randomly positioned. You would have to keyframe the layers before you apply the script, then apply it and keyframe their position ahead of the previous keyframes.
I personally would just manually keyframe everything in this instance, just because I feel like the amount of tweaking you would end up doing using another method would add up to just as much work. Who knows though; I've been wrong before. ;)