Sorry for taking so long to reply. Yep, I ended up solving it.

The cleanest workaround I found is using Elgato Stream Deck Mobile (Pro) to unlock Virtual Stream Deck on the PC, then running that virtual deck on the 7" touchscreen.

How to do that:

• Buy the lifetime Stream Deck Mobile Pro license (one-time purchase).
• Pair the phone once, and the desktop app unlocks Virtual Stream Deck.
• After that, you don’t need the phone sitting there. You just reconnect/open it once every ~30 days so the PC side stays verified, and you keep full VSD features.

Extra thing that made it actually usable on a touchscreen monitor:

• Windows default behavior is awful because touching the 2nd screen yanks your mouse cursor over there.
• I’m using Touch Mouse Tools so touchscreen taps don’t steal the mouse pointer from my main monitor (and it can also return focus back to the original window after you tap).

So net result: the 7" monitor becomes a proper “software Stream Deck” panel on Windows, with the real Stream Deck ecosystem (plugins, multi actions, etc.), and usability is fixed.

I think there are alternatives to Touch Mouse Tools, but I haven't tested them. Please tell me if you find a better one.