Firstly, you can render to a target backbuffer without a swap chain. That's possible, the issue you really have is basically taking over a section of the users desktop to have priority on the desktop. I believe the ability to do this removed back in win 98 days (someone can confirm).
So really, you are left with doing annoying things to bring this to the front. That being, you either can use a windows form or a UWP, but they are in a window. You can though remove the frame of the form firstly to make it look like it has no parent window (UWP does this well). But then you are left with then bringing it to the front of everything, you can do that with a thread that constantly checks if it is in focus and bring it in focus/to the front. But this has the issue that it then takes focus off other applications which makes it aggravationware. I haven't seen other methods that can simulate your requirements, but that's the way I would look to solve for it (for now!).