To add to the options for phone purchase registering with the CoG site, there’s an Omnibus app available for Android via GooglePlay that if you restore the game into your Omnibus library or purchase it in the Omnibus, it’ll register the game as purchased on CoG site too. For Hosted Games, anyway. Don’t think the CoG games android omnibus is released yet.
Almost all of my HG titles were purchased before I figured out the “view site as website” option in my phone browser, and they are all available via the site now, without staff intervention.