If you copy the NBT data, take the base64 encoded in that and decode it then go to the textures.minecraft.net link, that's the file.
Looks like pillow has given a solution here! Will close now. (PM Pillow with any further questions as they seem like the knowledgable type on this! ; ) )
Thank you for your thread on a recent query! It seems like it has now been resolved. This thread has been locked moved to Help Archive.