Nevermind. It was just a case of the mornings. I typed in the name of the wrong post type. Anyway, if anyone is looking for the same answer, the url is just …
http://www.example.com/api/posts/update_posts/&post_type=POST-TYPE-NAME&id=ID
I just upgraded a WordPress network to 3.5.2 and subsequently lost the options entirely to Set Featured Image. I deactivated all plugins–I don’t use many …–and went through one-by-one see if a plugin was butting heads with something else, and the culprit in this case [for the first time ever] was Adminimize 1.8.4.
Let me know what other info I can provide to help troubleshoot.