1W is a unicorn as just plugging in a power supply with no PC parts hooked up will register 1W. <10W is more realistic. Select a PC that can run off a laptop charger. Check manufacturer spec sheets for the idle power consumption. Don't install any PCIe cards or hard drives. Use powertop --autotune.

> Select a PC that can run off a laptop charger.

(looking at my 330W laptop charger)

I’m not sure that is the constraint you think it is.