After further investigation, it seems to be related to the internal webkit engine used by AIR (maybe a specific user agent that is not properly handled by Twitter?). You can provide an extra parameter to your StageWebView (set to TRUE), that will then force your app to use the native browser. For me it then uses the IE11 engine, and it works (the CSS doesn’t seem to be loaded though, giving an ugly login page).
It would be much better if the Twitter team was providing a fix though, so we can use the native webkit version of AIR.