If your deer is still a fawn, you will need to wait until it grows (in one month) before pelts stick permanently.
After you get a set, open your menu (escape key or the green dot on your action bar) and go to Network > Save current deer appearance, and you will be able to load whatever set you save at any time. Make sure you load it before logging out, however, as whatever you are wearing when you exit the game will be what you log in with next time.