how might one spoof geographic location on a desktop Mac

Is this site web based? Or app based?

If this is a web based service one idea that might work is to change your user agent in your web browser. For example, in Safari there is Develop menu that you can activate via the Advanced tab under Preferences:

Once activated, go to to the Develop menu and choose the User Agent option. There is a nice preset list of common user agents. So in this case just change it to one of the mobile device options like Safari iOS7 — iPhone:

Now with that done, go to the site in question, login & see how the site reacts. Mind you this is simply a brainstorming idea of a tact to spoof a mobile device. For all you—or we—know the site in question does server based geolocation based on IP address or just reacts differently from a real mobile device. And if that is the case, chance are you are out of luck.

But the choice of how/when geolocation data is added to a site is really based on the site owner’s preference in most cases and not controllable by end user interaction.