Did you try setting the user agent string to the latest version? I don't know how the site detects the old browser version, but if it's done via the user agent string, it could be changed. If you want to give it a try, google how to do it (there are probably add-ons available that can do it).
That said, I think you should find a way to disable WebP images, if that's possible. The latest version fixed a security bug related to such images, and maybe that's why Twitch thinks it is their duty to force you to update.