https://fwupd.org would be the way.
https://github.com/fwupd/fwupd

edit: Read your cross post. Using an USB would be the way then, or booting Windows.

as a newcomer i would just suggest getting a USB storage of some kind as they are cheap and the alternatives are not at all easy or obvious.

if you really want to do it without one, you could create a new fat32 partition and install freedos on it and dual boot to that and use that to run the bios updater-- assuming they have one for dos and not only a windows version, otherwise you'd need to do that but with an actual windows install. Modern windows doesn't require a license so you could just get win10 or 11.

but the act of resizing your disks or trying to reconfigure your bootloader(and especially installing windows) are all things that can easily result in you breaking your Linux install, likely irrecoverably without even more in depth work. so really..only do this if you're at least okay with the idea of reinstalling Linux and losing all of your data and spending a lot of time learning more than you might want to about how these things work.

so definitely easier to just get a cheap USB drive.

I haven't tried this, but if you are required to run an exe then you could try it from a Windows preinstall environment.

Hiren's is pretty well respected https://www.hirensbootcd.org/

Ymmv.