How can I make my CLM environment accessible through the public internet?

whatever 'name' you use to access the server, the 'publicURI' must be defined on the internet to get to your outside address, and somehow get thru the firewall to the machine where the server is running.. (this is where Abraham mentioned talking to you networking team)

I do this on my personal systems. I have a dyndns.org name, and I map it to my internet connection. then I use my router to map the port for the server to the machine & port on my local network which has Jazz running. (I have multiple versions running concurrently on different ports so that I can try.demonstrate different features) at the same time.
all the machine on my local lan, have a hosts entry that points the dyndns.org name to the same server, so that the pages returned from RTC (which have the public URI encoded in them) will work properly.

you need to talk to your network admin to see what is possible in your organization.