Since you've logged in to most of the browsers then I think it's time for you to test and use their app and maybe that will change the result. I've just also read it somewhere years ago and tried that solution and it worked, hopefully it will work to you.
Did the app bypass the IP verification procedure? If it doesn't then I don't think it can solve OP's problem. OP can't log in because he uses a new IP, so the client he uses doesn't really matter afaik. CMIIW.
I also sent them +10 DM on twitter...
That's really weird. Have you tried mentioning them in one of your tweets before you sent a DM? Maybe they never check DMs unless you ping them first. But if you don't have a ticket ID, I think it will be a bit difficult for them to follow up since they always ask for it. It's horrible customer support for sure. I also suggest checking the Reddit account posted above. They don't seem that active but they respond in a timely manner if there's a new thread about them.