Adding the specific Domain-based groups back in through Web Client worked, but I have to re-add ALL of my custom users, groups and permissions, now. Not happy about this, but at least I got it working again. Once again, VMware has released another vCenter mess. I really miss the days of 5.0 double-click vCenter that took 5 minutes to build.
Anyway, try this article, if anyone is still stuck:
http://blog.asvignesh.in/vmware-vcenter-5-5-you-do-not-have-permission-to-login-to-the-server/