NVR Screen on/off by cabinet door?

I'm not sure if what I want exists, but this seemed like a good place to ask. I'm still fairly new to security cameras.

I have a Reolink NVR 36 in a well-ventilated PAX wardrobe with the rest of my networking/home server equipment. Inside the cabinet is a monitor for the cameras. Since the door is usually closed, having the screen for the NVR on all the time seems like a big waste of power - I'd say I look at it once or twice a day on average - basically just if I'm near it and hear something that could be outside.

Ideally, I'd have a monitor with instant-on tied to a door sensor, but if I do that with the power supply, I'm not sure how long it would take to load - I also wondered about somehow connecting a KVM to the door instead so the monitor would just go to sleep from lack of signal. I don't want to spend huge amounts of money on some business display screen with presence detection, but I am willing to jump through some hoops and over-complicate things to get a screen that comes on as close to instantly as possible while cutting down on power use, heat generation, etc.

Any ideas or anyone who has done something similar?

Note: in case any recommendations are super specific monitors - being in the wardrobe limits me to about an 18in width.