User cannot see project custom fields despite having the security permission Allow access to custom fields enabled on the security profile (Settings > Access & Permissions > Security Permissions).
The reason for this is that even though this security permission was enabled, the Security Permissions > Custom Field section was not assigned to you. BQE CORE has very granular security permissions for unique situations and so it has to be allowed at two levels in this case. Granting permission to this screen will resolve the issue.