I just figured it out! I noticed under GitLab's Settings -> Applications -> GitLab Mattermost, it had an OAuth callback URL that wasn't correct (it had the wrong external base URL in it, probably from an earlier time when I was trying to configure Mattermost properly). I corrected it to the proper callback URL and things are up and running.
I would think that GitLab should set this value up automatically if I update the Mattermost external_url in gitlab.rb and then do a gitlab-ctl reconfigure, but I may be mistaken.