I did not try your method yet, however I debugged the problem in my method. It's something (I guess Lomiri itself) that is sending a signal when the window is paused while switching. It is killing xev. The fix is to use TweakUT to keep the terminal awake, then my method works whatever the launch mode (direct > or with starting a new task &>)
So I could repro the problem you have seen. No discrepancy here.
I have looked a bit at Firefox and Chrome under vanilla Linux (Kubuntu 24.04) and xev reports something very different than under UT. That is, no strange events, basically the system reports the menu being created and mapped and that's all.
I have also 'packaged' a recent version of Firefox freshly downloaded from the editor's site, that is, I did
env GDK_DPI_SCALE=2 MOZ_ENABLE_WAYLAND=0 DISABLE_WAYLAND=1 ./firefox
and the menus in the top bar display (if no scaling were done, I'd guess that the menu would display if by a stroke of luck I could hit the minuscule screen area)
So my guess is that's not a clever packaging trick, that's something that Firefox is doing differently than Chrome. I searched for a Chrome Linux arm64 build but it seems that Google has insufficient resources to generate arm64 Linux builds (it's not a joke, it's exactly what is said by a Google employee in the Chrome issue tracker).
I'd say that as under UT Chrome and Firefox are hitting XWayland that is hitting Mir, the problem could be all the way down to Mir. The version of Mir in current 24.04 UT is woefully outdated. When it will be updated, the way forward will be to switch to Wayland anyway and drop XWayland.