VR resolution vs screen resolution

[FFlow]

Just bought a Valve Index and I would like to play an "old" sim using DX9 with Virtual Desktop, like I have seen in numerous tutorials. The sim in question is Falcon BMS.

So the Index has a resolution of 1440 x 1600 / eye. My computer screen has a resolution of 3840 x 1080.

If I run the sim in windowed mode at max res on my screen, it runs at 3840 x 1080 pixels.
Using Virtual Desktop to display this on the HMD would mean that I run a lower resolution for BMS than what the headset is capable of ? Or is the resolution of the desktop in Virtual Desktop completely independent ?

I'm not sure I am looking at this the right way, but in the end my goal is to display the game in the maximum resolution / quality my headset can offer, so if you have any input for me it would be highly appreciated :)

[ggodin]

Hi, the effective resolution of the Index when looking at a virtual screen is around 1920x1080 so playing a desktop game would feel be slightly lower compared your physical monitor (but still look great). This is because the virtual screen doesn’t take the entire field of view and there’s also loss of quality during distortion.

[FFlow]

Thank you for your answer. Goal is actually to have the VR desktop in "full screen" and use an external tracking solution to handle the head tracking in the game.
See this video : https://www.youtube.com/watch?v=5_q7nEDSSsE
So I would like to run the game in the best resolution possible for a good result.
Would it be a good idea to add another "high resolution" screen to have the game run at for instance 4K and then display that on the HMD ? I could use a dummy HDMI plug to achieve that, like : https://www.amazon.co.uk/Headless-Display-Emulator-Headless-1920x1080-generation-HDMI-2K/dp/B072PV6K8J
Would this be a good idea ?
Thank you :)