Google CSE Integration with Varying Countries

Hi Team, my answers experience is experiencing difficulties with their Google CSE integration (Links) - this experience has a global website structure with individual pages for different regions (United States, Europe). The issue is that often in the links section a link result for Japan will appear for someone who is on the US site - when clicked on this link will take them to the Japan page and change their country location. If the user’s country location changes on the user’s site, then the header/user experience changes on the company pages.
Originally, we were going to create different search verticals for each country, but because of their set up (not a true search bar integration, clicking an icon takes you to a blank search page) we wouldn’t be able to use Query Parameters / Context and show the appropriate links vertical. For example, you can see there is no “referrerPageURL” when you navigate to Answers from the homepage:
The other option is creating 21 individual answers experiences for each country, which is cumbersome . I am working with the client to find a solution on their end, but I wanted to open this up to your team to see if I had any other options or if you had any thoughts.