So it turns out pressing F12 during boot has an option “BIOS Update” which sees the BIOS .exe file on my USB device. So, the answer is no, Windows is not necessary, unlike what it says on Dell’s website.
Of course I found this out only after downloading FreeDOS (twice! “USB” option is a VM image…). “Restoring” the ISO to the USB drive is great, but then I realized I couldn’t add the BIOS file to it. I also tried creating two partitions on the USB drive, the first one to restore the FreeDOS ISO to, and the second one to contain the BIOS update file, but it failed to boot. So maybe that’s what you meant by “a bit hairy to get there”…