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Most operating systems let you choose the output device of the audio.
So, you should be able to plug the headphones into the computer itself, then in software choose to have the audio go out of the HDMI when you want the monitor speaker, and line-out/headphone when you want the headphones.
Which distro are you using?
I had forgotten that there is an audio port on the pc itself. I guess that would be the easiest solution, I'll just have to buy a longer cable.
It's not hard to switch sound devices if you leave it plugged in. If you're on Windows just right click the volume icon on the taskbar to the right and pick the output you want.
no need to even right click, on win10 you can click it, and then just above the volume slider you'll see what device you're using currently, if you click that you'll get a menu where you can pick from all your available outputs
The monitor may have a menu with an audio option to switch between them.
You could get a small usb soundcard. You can get cheap ones on Amazon.
Then, just swap your audio destination in your OS from your monitor to your soundcard when you want to use headphones
If you already have an Xbox controller with a 3.5 jack you can use that too as long as it's wired, on Windows at least
Also works through the proprietary wireless dongle. Doesn't work if the controller is connected via bluetooth though I think
You can use a program like VoiceMeeter. You set up your inputs and outputs, then can change where your audio is routed as needed from your computer.
I think the 3.5mm is hard wired to override the monitor speakers
You could by an extension cord.
This is one reason I use computer speakers. Based on what you said, quality doesn't matter but most of them will have a headphone connection so you can just plug into them. I much prefer high quality, hence why I chose the ones I did, but it will allow for easier plugging with potential, almost guaranteed, higher quality than the monitor audio.