Alright, so it turns out I already had the HWE kernel. Installing the AMD drivers off the site seemed to have no effect.
I switched back to my non-graphics module shell and that worked for a while. I did have one login freeze, but overall much less often. No mid-session stuttering.
Since I actually do want to use the $500 module I paid for, I’ve since in-place upgraded to Ubuntu 24.04 and have been using it for about a week. The good news is the login doesn’t freeze anymore, and 0 mid-session stuttering. The bad news is…touchpad doesn’t work 2/3 of Ubuntu boots, the JetBrains AppImage launcher tries to install itself on every boot (unsuccessfully), and the Steam application has trouble showing itself most of the time. Still, those are persistent annoyances rather than major blockers. Still, would be nice to get my touchpad back (it’s not an uncommon issue apparently).
So my take is…while the Framework 16 has official support for Ubuntu 22.04, the Graphics Module does not work well and needs Ubuntu 24.04.