The BIOS update instructions seem to require a functional windows installation which I currently do not have due to a problem I am trying to solve. Is there a way to upgrade the BIOS through the BIOS itself?
You only need an OS that you can boot into and use to create an EFI based usb stick.
No need to use Windows. You can for example boot into Ubuntu from the LiveISO and create your firmware-upgrade stick from there.