Loading a Random Resource in a Folder Without Using Resources.LoadAll

I'm working on a game designed for use on an iPad 1, and I'm running into memory problems when trying to find a random texture. Basically, I need to pick a Texture at random, but when I try to use Resources.LoadAll, it exhausts the device's memory and crashes. I need to find a way to look at the file structure of the Resources folder without actually loading anything.

In other words, I need to find a random file name within a given folder, not a random file. However, I don't see any function in the documentation for Resources that looks like it could help me, and the files in Resources are compressed when compiled for the iOS.