Not sure if anyone is still following this, but I wrote a small, single executable .NET program (for Windows only, tested only on Windows 10 x64) called Magic Window. It's essentially a useless, function-less Window invisible to us, but not OBS. OBS will capture the contents of the window, but we are free to interact with anything and everything else.
I say "us" because I made it for myself to solve this problem. Potentially others may benefit from it, so they can find it here on this thread as well. I will only continue to develop it if there's incentive to.
You have to configure OBS to capture a Window and configure it not to capture the cursor. It kinda sucks that I'm not able to easily capture the entire display's composition including the mouse, but I get around this by drawing a cursor onto the invisible form.
The behavior is that if the bounds of the "magic window" is outside the primary display, it forces itself back in. Expect the cursor to always be drawn in the center IF the window is within the bounds of the display entirely. There's no "gliding", OBS will see a 16:9 image painted onto a Windows form that you cannot see/nor interact with.
By default, this window is sized to 854x480 and paints a zoomed-in image 24 times a second, but the size and FPS are configurable by passing command line arguments, like:
MagicWindow.exe -width 1280 -height 720 -fps 60
Optionally, in the system tray (the only user-friendly way to kill the app) you can select preset resolutions and/or halve the current form's size, or double it. Tested on Windows 10 only. I assume that it will work on any machine with .NET 4.5 installed, but it might not work on stuff like Windows RT because this .NET app does rely on system calls to set window features that cannot be done in .NET.