disable or move keyboard-switching dialog while leaving functionality intact

Question

I have several keyboard/language layouts on my Windows 10 machine, and I frequently switch between them using Windows key + Shift. My issue is that when I'm sharing my screen in a meeting, every time I switch languages, the keyboard-switching window shown here comes up:

This can be distracting, especially to other people. Is there a way to continue to be able to switch layouts but suppress the keyboard-switching window, or control where it appears so I can move it to my auxiliary monitor (my laptop monitor), so that it won't be seen when I share my main monitor? I'm okay with the "language bar" display in my taskbar changing, just not the larger dialog.

Note that I do not define "hot keys for input languages" in the "Text Services and Input Languages" dialog (since this has never worked correctly for me, and since I would rather use Windows key + Shift rather than Ctrl-0, Ctrl-1, etc.). Thus, the key sequences for these are deliberately set to "None":

Answer 1

I used the approach suggested by @JosefZ in his comment, but I had to take some steps that were complicated enough that I wanted to mention them together with screenshots.

Here's a screenshot that shows the "Change Key Sequence" button (at the bottom); clicking on it brings up the dialog box that covers the one behind it.

Note that it's possible to add a sequence that changes the language, and/or a sequence that changes the layout, but you can't do a single cycle through all language/keyboard combinations as you can with the Win + space keys. This is actually fine with me, since I generally want to cycle through languages. (For English, I have a US and an International layout, but I usually only need to use the US.) None of the keys or key combinations worked for me other than the grave accent key, but that was also fine with me, even though it did mean that I had to come up with a new keyboard layout because I was already using a Grave + AltGr + Shift key combination.

Getting to the dialog that shows the "Change Key Sequence" button is surprisingly unstraightforward. Searching for it in the start menu didn't work. I had to go to "PC settings", then "Time & Language", then "Language", then "Input language hot keys":

I also recommend setting "Let me use a different input method for each app window". This makes things much simpler for me to work with, given that I often want to type in one language in one window and a different language in another. Now I don't have to change the language every time I go from one window to another.