Can I take the LFCS exam from Fedora or Arch?

Being able to run Linux locally doesn't necessarily mean the exam software and proctoring checks will accept it. I wouldn't assume either Fedora or Arch works just because it's Linux.

Check the current exam provider's system requirements and compatibility test, then ask support specifically about your distribution if it's not listed. A successful test is useful, but it doesn't necessarily make an unsupported setup officially supported.

There's a distinction here between your host machine and the environment where you do the exam tasks. One account in the discussion described being given Ubuntu virtual machines for assignments rather than choosing a distribution. That doesn't establish whether Fedora or Arch is supported as the host.

For studying, Fedora should be useful, and Arch can teach you plenty about Linux. For exam practice, though, I'd also work through the actual task types on a distribution close to the stated exam environment. Different defaults and tools can build different habits.

I'd have a supported machine or operating system available if possible. That avoids gambling on an unsupported setup on exam day, even if a preliminary check happens to pass on Fedora or Arch.