I don't know if some people still have trouble with this issue, but there are some news I want to tell you:
A few days after my last post in this thread, Line6 contacted me directly and offered me to send the guitar in and handle it as warranty, although my warranty time is already over (thanks Line6!).
Now I got my guitar back and the problem is solved. They simply changed the three way switch, which seemed to have been faulty. Now everything appears to work correctly :)
So again big thanks to Line6 and maybe this helps other users to solve that problem.
Benni2407's post in Jtv-59 Toggle Switch Fails After 2.0 Update was marked as the answer