When I use another Mac through Chrome Remote Desktop, the input source shortcut isn’t passed to the remote side. Every time I moved between Korean, Chinese Pinyin, and English, I had to find the menu and click it.
So I made a small bar that always stays in a corner of the screen. Pressing a button switches the input source right away, and the current input source is shown by color. I often took partial screenshots of the remote screen, so I added a button that puts one on the clipboard.
It only runs when the input source changes, so it uses almost no CPU. It is a single-file Swift app and can also be used from the menu bar.