So this project was discontinued after couple of weeks and I thought I should write why to help others that may pursue a similar task.
In order to be able to create something like a virtual desktop, a Windows API should be used. Newer versions of the Windows API offer lots of tools to ease desktop capturing and interactions, however same cannot be said for older ones (like WinRT).
And the way to go with Godot is to create a custom C++ Module, utilizing the Windows API. However, apart from difficulties of building Godot frequently with this newly developed module, there are C++ version incompatibilities. Godot uses C++03 unlike the modern Windows APIs which use the newer C++17 or above.
So keep this in mind, if you were to dive into such a project.