hm, that really nice solutions - what capture card do you have?
The capture card/dongle are designed for using in Twitch game streaming where you are using it to capture the output from a game and feed it into something like OBS. The one I have was pretty cheap, but only supports an output resolution of 1080p (input up to 4K). I don’t remember the exact brand, but the above should give you enough keywords to get started.
There are cheaper options if input lag isn’t an issue.