I honestly dont know of any computers that you cannot get into the BIOS unless it was locked out on purpose.
Simply resetting the Bios on the motherboard jumper will in most cases allow you to get back in as it would normally cause an error on boot up and offer you the bios option.
There was another guy somewhere also trying to do this with an Arduino but thats pure overkill.
Adding a GPU also may need that you enter BIOS first to tell it which GPU to use as primary.
Had the black screen here on some boxes and it was just a matter pf plugging the monitor into the old GPU port first to get into BIOS...Change GPU order then reboot with monitor plugged into the new GPU.
if you really must use an Arduino you may want to consider a relay board to isolate the Arduino from the computer and just use the relay contacts to trigger the power switch.