I got it running! (Haven’t tested it on the shapeoko yet, but the program starts and I get the GUI).
flatpak remote-add --if-not-exists flathub https://dl.flathub.org/repo/flathub.flatpakrepo
flatpak install flathub org.kde.Platform/x86_64/5.15-23.08
flatpak install CarbideMotion.flatpak
It then runs as expected with flatpak run com.carbide3d.CarbideMotion
Why it needed the KDE platform on the same version to be installed a second time from a different repo is unclear, could be that there was some missing dependency of it. Linux fails in mysterious ways. It seems a bit odd that Ubuntu didn’t have flathub installed by default as an available option.
This is the first I’ve run into problems with a flatpak (surprised it took this long given the long history of problems I had supporting desktop Linux at work which eventually led to us switching to Macs). It might be worth it to try building it as an appimage as well to see if one system is more reliable than the other.