I have three (FX, X-plorer, and the one built-in to Android Assistant), but none of them allows me to copy the path, at least not that I can find. I may have to check out Ghost Commander.
But I think the problem has to do with the task I'm using (Play Ringtone). The Search icon only directs me to a listing of sounds in, I guess, default folders. If I use Play Music, the Search opens up a file manager and I'm able to navigate to the specific file I want to use (the path displayed in this task is "ext_sd/Sounds/