Unfortunately, I don't really know what do you mean by this. PWAs use completely separate profiles, so it shouldn't matter which profile you use on your main Firefox. If you want to sync data between your main profile and PWAs, I recommend you use Firefox Account/Sync or a third-party sync solution.

If you mean that opening a link from PWA in a default browser opens your link in a wrong profile, then go to "about:profiles" in your main Firefox and set the correct profile as default.

If you think there is some bug or other problem, please open an issue/discussion on GitHub (and write it in English if possible).

I released an update that fixes compatibility with Firefox 101.0 yesterday, but it hasn't been approved by Mozilla yet. Until it's approved, you will have to manually download the native program from GitHub: https://github.com/filips123/PWAsForFirefox/releases/tag/v1.4.2

Flatpak (and Snap) currently cannot work because of sandboxing limitations (and the same is true for any extension that uses native messaging API). This cannot be fixed until native messaging is supported by Flatpak and Firefox. You will have to use normal Firefox.