Another suggestion would be to get a basic Windows 10 tablet, which range around $80 to $120 or £60 to 80 in the UK. Although they have bare minimal hardware (32GB SSD, 1GB RAM, etc.), they do run the full Windows 10 installation, so you can run most diagnostics as with a full size Windows PC, such as Auto test here without it stopping if the Window is put in the background. Teamviewer also works great for remote desktop access.
Most Windows 10 tablets are powered over USB also, so could be kept charged from the router's USB port if it supplies sufficient power. The internal battery also doubles up as a UPS, so a brief power interruption will not knock it out like a mains-only powered PC.
