You can use a older possibly partially broken laptop with low TDP CPU. Put the guts in a larger PC case or something similar. You can calculate what kinda heat sink you need that works without fan or pick a fan-less device. You still need a bit air flow so the case must be open enough or has fans depending on the CPU and size. The last time I did this was several years ago and It was running 24/7 for about 2 years without any problem. A Laptop alone likely would dies someday because the fan dies. On idle the my whole system inclusive power supply was burning less than 10 watts.

But there are far far more efficient CUPs noways. You basically have the last 10 years of mobile CPUs to find something that does exactly what you need. Some older CPUs you can get almost for free. If you need something newer (with 4k video HW rendering an stuff) you may get a good dead with broken stuff. especially display and keyboard can be broken since you wont need them. make sure it has the IO you need especially RJ45 are sometimes missing on the very slim new devices Be aware that some devices can not be turned on without keyboard attacked! Either because the bios doesn't like it or more likely because the power on button is on the keyboard so an USB keyboard wont work! Also auto boot after power-lose may not be possible because some bios just do not have that option. So make sure you know exactly what you need first.

DO NOT (AB)USE THE LAPTOP BATTERY AS UPS. The battery will die within months if its always connected and all you get is a potential for a house fire.