A common way for avatar theft to happen is by abusing the game's automatic content delivery system.
In order to mitigate avatar and asset theft from said system an implementation of a sort of "reverse whitelist" could be used. Essentially, it would function similar to how showing avatars would work, but in reverse. The individual user would have to allow other users to view their avatar fully, otherwise a fallback is used.
So seeing avatars would be a two way system: You allow X user to see your avatar and they show it\allow it via safety settings. This would potentially stop the aforementioned abuse of the content delivery system currently in use.
Perhaps there could be an option to individually allow users to see an avatar as well as some blanket ones to give a finer control to who can see your avatars (perhaps an allow instance\all\individual setting).
All in all, this would protect users and their uploaded content as well as give them a more direct control over who views it\has access to it.