Don't run IIS nor Moodle behind a proxy but do know that for Moodle settings for proxy to work the web server must be configured. (do run a proxy on apache such that another service running on another port on same server is redirected from what appears to be a subdirectory of the server. IIS has to have something similar.)
See:
The DNS entries for both internal/external ... internal mapping the private IP to the FQDN and external mapping the public IP to the same FQDN ... is required but that just directs traffic from browser to server. Once traffic gets to server the proxy config of the server takes control.
'spirit of sharing', Ken