We resolved this issue. There was a password mismatch between our internal DB and the WP user and therefore the Forum Member. 

Originally we thought that password needed the MD5 hash. So we hashed the passwords and imported them to our WP users. We realized quite a while later that things were still not working. A manual change of the user's password in the site, produced a very different hash than we were seeing from the passwords that were imported. We believe WordPress is using a version of the PHPass framework. But we did not want to mess up the password sync again. We wrote an API to sync the passwords to the WP site and let the site hash them however it wanted to. This corrected the issue and all authenticated users are now able to access and use the Forums according to their set permissions.

You just need to use WordPress-specific functions to generate password hashes, and everything will work correctly.