No way to do that. Even a screenshot of simulated 4K using DSR would not result in an actual 4K screenshot. Your monitor's physical limitation of 1080p will not allow it. This is because DSR super samples to a higher res internally, but samples it back down to the res of your display. Otherwise your display wouldn't even be able to show it.
Same thing with trying to upload 4K simulated video content to YouTube using DSR. YouTube will only see it as 1080p content. You can resize to 1440p, and it will look better than 1080p only due to YouTube using way more bitrate to convert 1440p content, but it will still be lacking true 1440p pixel content.
If you're planning to exhibit the content via a slide show on YouTube or something, I would advise limiting it to a 1440p resize. If you try to resize it any further, the lack of pixels is going to catch up to you.
I have a couple videos that demonstrate the difference between a 1080p upload, and one resized to 1440p, which prove YouTube's higher 1440p conversion bitrate more than makes up for the lost pixels.
1080p Upload
Resized 1440p Upload
Even just viewing the embedded videos without going fullscreen, you can easily see toward the end of the escape in the stolen car, that the first video suffers from pixelation. This is because YouTube literally uses 3 to 4 times the bitrate for 1440p conversion, even though it's less than 1 time more pixels than 1080p. This is also why they recommend 12,000 bitrate for 1080p videos, but 30,000 for 1440p.
So in doing this project, you need to consider more than just what's going on at your end, but also what conversion bitrates the hosting site you're using has, if you plan to use a hosting site.
If you plan on doing a slide show, I recommend Avidemux if you're OK with minimal effects. It will do fade in and fade out, and Lanczos3 resizing, but it is very streamlined with no bells and whistles. Being streamlined though means it's fast and stable at encoding. It does however have ABR (Average Bitrate) encoding, which is far superior to VBR (Variable Bitrate). I recommend Two-Pass ABR, and 30,000 bitrate, with the aforementioned 1440p resize using x264 and an MP4 mixer.
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