I just downloaded Palemoon to check and it seems the CAPTCHA straight up crashes. Once it crashes, reloading the page no longer shows the CAPTCHA so it did pass something at least. I tried another Cloudflare turnstile but the entire browser crashed on a segfault, and ever since the CAPTCHAs don't seem to come up again.
ChatGPT.com is normally quite useful for generating Cloudflare prompts, but that page doesn't seem to work in Palemoon regardless of prompts. What version browser engine does it use these days? Is it still based on Firefox?
For reference I grabbed the latest main branch of Ladybird and ran that, but Cloudflare isn't showing me any prompts for that either.