W3TC Full Site Delivery as Headless WordPress?

I’ve got a couple problems I’m trying to figure out. I’d really appreciate it if anyone can help with the following;

1. Can we use FSD for running completely headless?
   We are trying to keep our WP server inside a private network and publish the site to S3 with CloudFlare as CDN.
   Is this capable of working?
2. If so, how do we change the domain when publishing?
   If the private server were ‘wp-authoring.domain.com’, that should publish as ‘www.domain.com’
3. Can we edit the path when publishing? If so how?
   The authoring site is hosted on the webroot “wp-authoring.domain.com/*”, we need to publish to a nested S3 bucket “www.domain.com/public/*”
4. What is the minimum permission set necessary for the CloudFlare account?
   We have a complicated hosting arrangement, our domain is shared among a number of services. We can’t allow the W3TC plugin to control settings for the whole domain. We can configure the CloudFlare settings and Page Rules manually. Assuming that the plugin requires at least “Cache Purge” permission, is that sufficient to allow the plugin to do cache invalidation on individual resources?