In Registry, Computer\HKEY_CLASSES_ROOT\SystemFileAssociations\ added an .lge extension, just like the other extensions that have MediaInfo in the context menu. Copied the structure and settings.
MediaInfo appears now.
In ShellMenuView, it shows up with the other MediaInfo associated extensions.
Updated the imgur link with the registry details.
LE: MPC HC does not have the .lge extension in it's default file types to open list.
MediaInfo does not show up in the Set defaults by app page. MPC HC is listed there, but after clicked Manage, the .lge extension is not there.
In the end, you're still going to have to look at either how files are associated by file type and.or by protocol to determine how the files that are doing what you want are set up, then duplicate that for this extension. That's how context menus end up being built, and if Windows cannot automatically figure out how to do it, you have to.
Also, is there anything within the program that you want to appear in that context menu that allows you to choose the file types it opens? You could go from that angle instead.
And, in what I consider a personal dope-slap moment, you would probably be best off using the Set defaults by app link, same location as mentioned previously, which rolls aspects of file type and protocol together.
You need to associate that file type with the app you want to have appear in that context menu, and that app should be in the list presented by activating the above mentioned link. This will likely be the most direct method.
I have no doubt your registry hack worked, but it can be achieved via user settings as I specified in my most immediately previous message.
I do not ever recommend editing the registry to accomplish that which can be accomplished through the settings interface. There is too much potential for unintended consequences through disaster (for many users, anyway).