Seems 8bitdo only offers a firmware update tool designed for Windows or Mac on their website, with no option to download firmware files directly.
Seems a bit deceitful to me, since they advertise compatibility with Steam Deck
You could try asking if the manufacturer can could provide the files, or see if the tool can work under Wine
Download that ISO and put it on a USB stick with ventoy, then boot up from it, download the Windows tool for flashing the firmware into the live environment, and then use it from there.
ventoy is in the repository. It allows you to simply drag & drop ISOs onto the partition on the stick, and it’ll give you a boot menu with each of the ISOs.