You've already received good advice in this thread. Make a portable windows install, boot into it, and update the BIOS with the vendor's windows utility.
If it can't be found and installed through fwupd and the maufacturer does not provide a file to install firectly, your next best bet is to create a portable windows installation with something like rufus and install it from there. Or contact their customer support.
So, you are suggesting to use Windows just to update BIOS. ( Asking for clarification )
Also not exactly sure how that would work unless you could download the update in a browser and it is executable in Windows as an exe and not coming from Windows Update.
They don't even provide BIOS file in the first place.
WTF that's strange. I would have said that BIOS updates should be mainly OS agnostic, but since Acer recommends a windows update to do a BIOS update that throws a wrench in things.