No, the Tweet Button won’t help you create a service that authorizes an end user and then uploads a profile banner on their behalf.
Creating a service like this will require a bit of work. Is there a programming language that you prefer over others? You will need to do some server-side programming to accomplish this no matter what. You’ll need to learn how to perform the OAuth token negotiation “dance” and then you’ll need to learn to perform a multipart POST to upload the image on behalf of the user. More on: [node:3240].
The “button” part of this is the simplest part of it all but also the last thing you need to figure out how to accomplish.