Drupal - Upload an image to filesystem, rename it and set it as Profile Picture for current user

I agree with @Clive you can do it that why, and also you can download the image directly with the full_name.
To do so just change the line:

$profile_pic = system_retrieve_file($picture_url, "public://profile-pictures", TRUE, FILE_EXISTS_REPLACE);

by

$profile_pic = system_retrieve_file($picture_url, "public://profile-pictures/$full_name.jpg", TRUE, FILE_EXISTS_REPLACE);

and remove the line:

$picture->setFilename("$full_name.jpg");