If caching is enabled, disable it for starters. If you see different IPs in the Joomla backend (remote vs local), then you're probably using some proxy which does additional caching as well.

Instead of relying on IPs, you should either use a combination of fake domains plus /etc/hosts overrides or perhaps simpler, just use xip.io. Or, even better, just create a subdomain on your domain's registrar that points to your external IP, e.g. office.seacrestcockers.com. Calling the domain externally, will always map to your external IP and as such serve your site (as you do with the IP already).

When you're working locally, simply edit your /etc/hosts file and add a pointer of the *local* IP to the same subdomain. E.g.:

192.168.1.100 office.seacrestcockers.com

This way, when you type "office.seacrestcockers.com" in your browser, your PC will resolve to the *internal* IP.

And this is how you get a common way to reach your site, regardless of your location.