Set up pc for two separate rooms optimally

Hey everyone. I play a mixture of competitive and casual games, and would like to use my tv in the living room to play casual games on instead of the monitor in my gaming room. Luckily, the living room tv and my pc are on opposite sides of the same wall. I installed two brush plates and ran an HDMI cord through the wall to the tv (10 ft cable length).

My question is, is there a simple way to make it where I can play my PC on either my monitor or my tv depending on what I am doing? I am currently using windows P and setting the project to secondary screen only. Really, I just want the computer to ignore whatever screen I am not using at the moment.

Because my wife wants to watch me play Resident Evil 9, I brought the pc into the living room the other day and it worked flawlessly with zero stutters, lag, etc.. But now that it is in the other room, I have gotten a few stutters. I am not sure if it has anything to do with it being on a secondary screen. Any help is appreciated.