"View full site" via link and $_GET parameter

I have a handful of client sites that are using the User Agent Groups feature to serve a mobile theme to mobile users. However, they also have a “View full site” link in the footer, and they want to serve the desktop theme when a user clicks that. The link presently adds a $_GET parameter that we can check for, but I’m not sure how to actually get the desktop version served. I have tried deleting the w3tc_referrer cookie when that’s set, but that didn’t work.

Anyone have any thoughts or perhaps a better route to take? Would love to avoid a 2nd plugin just for theme switching when w3tc already offers similar functionality.

https://www.ads-software.com/plugins/w3-total-cache/