Hi,
Thank you very much for the details, it helps greatly!
My guess is that category ID conflicts might be happening here. I think the best course of action would be to make another search bar, and use the first on the english and the second on the german version.
On the back-end, you can switch languages – so you can configure each search bar in the current language, that should avoid any conflicts. Then use each on the corresponding language.
Let me know if that changes anything!