I think I finally resolved it. I deleted Twitter off my phone, rebooted, waited about 24 hours, rebooted again, then reinstalled it. It hasn't logged off since.
There is something weird about how Twitter doesn't delete all of its info when you delete the app. I can tell this because despite deleting the app (and supposedly all app info), when I reinstalled it the notifications badge came up with the number of alerts I have despite not even logging in yet. When I just deleted and reinstalled immediately, I'd still get the logout behavior. When I waited 24 hours, the notification badge still showed up before logging in, but I haven't had it logout since.
YMMV, but that's what finally worked for me.