Can you just remove the tabs altogether? I don't care if I can't move or edit chat windows in game with the mods.
In this UI I'm going to embrace features like this, but focus on keeping the minimal/functional nature of the UI. But if you want Tabless Chat put these files from Finamdar in your Vert directory. They seem to work just fine. I would encourage everyone to try it as is first though.
(Note: I didn't need any of the *.cur files)