I took a quick look at vBulletin's official website, but I think I'm going to take a deeper look into it and see what I can find so I can at least bring something to the table. lol It looks as though this is an older version of vBulletin, though, right?
For discord, you would host the chat separate from the site, and it gives you a full suite of controls over moderation. You would just have people use their profile names from here so we'd know who each other is. It's very customize-able on the admin end from what I've seen. Just an idea. Who knows. In a year, they may come out with something even better--technology is always moving quick.
Thanks for letting me vent, though.