I agree with removing it altogether from the Control Center. It doesn't really serve any purpose there and just muddies the waters. The default region should be set under All Settings and then the per search region can be changed right at the search page.
As for how long a change at the search page region should last: I think that any time you search via the address bar (when Kagi is the default search engine) the default region from All Settings should be used. Even if that tab had previously been changed to another region. So, if that rules out option B, then I guess option A. But I would actually prefer option B where a tab stays in that region until it is closed, except if a search is made from the address bar.