SharePoint 2013 Project Server with Project Application connection issue

Are you using APM, or strictly LTM for this deployment?

If you use the HOSTS file, does this bypass the firewall policies as well?

What exactly is the issue, Client Integration? What you are probably seeing, is that while Internet Explorer is open, its maintaining a authenticated cookie to the site. When it is closed, the cookie does not exist, and the ports and protocols used by the client are unable to access 443.

Have you attempted to get a wire capture of the traffic? What is the difference in traffic when using the HOST file versus going through the F5?