Well, here's my contribution to this experiment...
This is using Rit's "Sunshine Orange" dye. I simply dumped the whole 1-1/8 oz box into a large pot with about 1-1/2 gallons of water for 15 minutes on medium heat, stirring occasionally (add salt and pepper to taste).
The cover turned out very... uhh... orange. I'm happy with the results. However, there is one caveat I should mention. I've yet to determine the reason, but as you can (hopefully) see in this second picture below, something during the process caused some blackening around the inside perimeter of the lens cover that I can't clean up.
This discoloration doesn't seem to be visible on any photos I've taken with it since though. So unless you really stare at the back lens cover, it really won't be an issue.
Good luck to everyone else willing to make the small sacrifice that is, their battery cover. I'm off to paint some happy trees and sheds. Bob... out.