MiddleClick-BigSur VS squirrel

MiddleClick - middle click with 3-fingers click/tap.

RIME/Squirrel - super flexible input engine for Chinese typing.

I don't know how to solve your problem but one open source alternative that basically works on every system is Rime. Seems like this is the MacOS one: https://github.com/rime/squirrel

Use RIME.

It's an open source IME for your computer, https://rime.im/ and it has support for a lot of cool input methods including pinyin and zhuyin of course, as well as Shanghainese and Cantonese phonetic input, wubihua, cangjie, and a whole lot more. You can also easily switch between traditional and simplified characters with a few keystrokes. My favorite feature might be that one of the pinyin input methods, 地球拼音, allows you to enter tones on your pinyin to narrow down the characters that it suggests.

If you're stuck only using stock software (like on a company PC), do this, otherwise I'd really suggested RIME. It's open source, free and supports a lot more types of Pinyin. Traditional Chinese included. You can even select Terra Pinyin if you want to be able to filter the suggestions by tone.

You can try rime.

Try Rime and choose "朙月拼音•臺灣正體", it supports typing traditional Chinese in Pinyin and fits Taiwan standard for characters.

As a native chinese speaker, I recommend you install rime. It's smart and basically support all kinds of chinese input methods. To install, refer to the ubuntu section of this page and install all the require package using apt. Then open the setting app, go to region & language and click manage installed language. First change the input method framework back to ibus in the "keyboard input method system" section. Then click manage installed language, select Installed/Remove language, tick the language you want to use (In your case japanese and chinese).