I did a quick search on the Bazzite github and found this issue. A comment suggested that you may have to set Fcitx5 to autostart (I believe you can refer to the Fcitx5 documentation on this).
It’s been about 2 years since I last used Fcitx5, so I don’t remember much, but could it be that you have to switch between character modes first? For example, I’m using ibus with Anthy and while I can switch using the system tray menu, I can also do it using my preferred shortcut of Ctrl+\. That is also the case with ibus with Mozc. Plus, Mozc assigns character-switching to the special JIS keys that don’t exist in US-ANSI keyboards (Eisu, Muhenkan, Henkan, Kana…), which is potentially why it’s not working for you.