I had to tackle this problem as well for a small project and found that I could just send a GET request to https://www.cloudflare.com/cdn-cgi/trace and parse the response with a simple regex: /loc=(\w*)$/gm The response time averages 10ms and is free, granted I'm not sure if there is a ToS or anything attached to this endpoint, I've only found non-conclusive discussion here https://community.cloudflare.com/t/what-are…
I'd advise you to just create a Cloudflare Worker if you want do this. You'll get more reliability etc. and Workers have a very generous free tier.