I am able to use the XML API to set a backlog value on a case. The following example works to make a case the top item in the backlog:
http://localhost/fogbugz/api.asp?token=[secret]&cmd=edit&ixBug=13984&plugin_projectbacklog@fogcreek.com_ibacklog=1
How can I submit that a case be removed from the backlog? Ideally, it would automatically include the "Removed from backlog" in the sChanges column and renumber all the other cases.