Host’s home page caching + Polylang

I’m having big troubles with the “Detect browser language” automatically showing the correct version (DE) to site visitors in Austria (who have their browser set to German), meaning it’s not. Then, when they switch to German, the cookie doesn’t seem to stick correctly. The problem is that the host (WP Engine) has strong home page caching and it’s policy to not allow it to be adjusted. So, the plugin combined with my host is basically not working. I’ve gotten good support from the plugin author, but question: Will changing to another host who allows control of homepage caching (assuming there is one?) truly fix this? Is there some other “detect browser’s language” solution I’m not aware of?

The page I need help with: [log in to see the link]