The only call that I think _might_ be dangerous to expose is the login API, but that should be rate-limited and / or have lockout anyway.
I've opened a bug to support CORS in FathomCloud. It's not a lot of work (I already have the filter in the repo, it's just not configured in). I just want to think it through to make sure it's safe: https://github.com/fathomdb/cloud/issues/50