@simon and I worked privately to resolve the issue and wanted to share with everyone in case anyone else has the problem.
The issue was with the ‘Publishing Username’ in the WP Discourse plugin (see: http://share.ptlrdsgn.com/3d2P230G1Y0C). We had customized that to a different user, but our user was not an admin in Discourse. Any user you use in this field must be an admin in Discourse. We fixed that and the logouts sync as expected.