22.04 / jammy here.
I'm still using the mate-dock applet as I prefer it over plank in Ubuntu Mate (yes full Mutiny desktop).
I recently switched over to flatpak firefox because of some loading speed issues (not start up, but a particular page resource I use often for work that was really slow with snap, otherwise it was fine).
I was able to get the flatpack icon to pin / stick on the dock after a logout by first pinning it normally to the dock then opening Caja with sudo, and copying the flatpack desktop config file over to where the dock expects to find the standard firefox desktop config file.
The dock expects the deb/snap desktop config file to be located at: /usr/share/applications folder.
Flatpack's desktop standard config file location is: /var/lib/flatpak/app/org.mozilla.firefox/current/SOMEHUGEIDNUMBERFOLDER/export/bin folder.
After a logout and reboot test, the dock's firefox icon remains and functions as expected, just like the old deb and snap versions did.