I think you are in good hands with the assistance of @dhowett.
I’ll mention that I think people have had issues installing very recent bios updates on very old versions and stepping through some intermediate versions was successful. There are differences using the Windows installer, lvfs and efiboot methods.
Research that if you don’t have luck with installing 3.19 directly on top of 3.06
Also there have been issues with the battery being fully charged with some update versions and installers. It may be useful to drain your battery 30-40% and then connect the power adapter just before beginning the update. The idea is to keep the battery charging through the whole process but also have the batter provide full power for short times while the update impacts some components on the system board. This is mentioned in the release notes for older update versions: https://knowledgebase.frame.work/framework-laptop-bios-releases-S1dMQt6F
There is more in the forum. I know it can be a nuisance looking for this info in the forum. Sorry that I don’t have handy links for you.
EDIT: Check the replies in this thread for one instance: Framework 11gen Linux - BIOS 3.20 error
Good luck!