As you have discovered, you cannot do this with the current API.
See, and upvote, this related feature request: Is there an API to upload images to SE's imgur installation?
See, also, Using the Stack Exchange API to update my user profile? on Meta Stack Exchange.
Your only current possibility is to manipulate the user's normal profile page, if you can.
I doubt that this feature is blocked by "constraints imposed to prevent abuse". It is more likely demand (low) versus difficulty (medium to high).
