It was working at some point, by putting the binary under ~/.var/app/org.mozilla.firefox, and running a few flatpak override commands, but now it doesn't work any more.
The commands were
flatpak override --user org.mozilla.firefox --own-name=net.diegoveralli.tabreport
flatpak override --user org.mozilla.firefox --filesystem="$HOME"/.var/app/org.mozilla.firefox/.mozilla/native-messaging-hosts/:ro
Firefox might have tightened up the sandbox since then.