When reporting an issue, please try to reproduce the issue with just Dominos and Dominos_Config running. Dominos reuses a good chunk of the stock UI, and other popular addons, like Move Anything, like to take control of those parts, too.

Addon Version
11.1.1.-cata

World of Warcraft Version
Classic Cata Live (4.4.2.59734)

I was hesitant to classify this as a bug since it is not reproducible with just Dominos but under this

Any other addon that works on Blizzard UI for action buttons should also work with Dominos, too. This includes things like

maybe it is worth a look (no expectations 😊 ) Apologies if it should be a feature request/enhancement

So this little addon by Gello FlyoutBindings (maintained/recent version can only be found on wowi) does override(?) bindings on flyout menu items.
The way it works on

• Default / Blizzard Actionbars
• Dominos reused Actionbars (ActionButtonN buttons)

Click or key the "root" menu button > flyout pops up, menu items re-bound to 1-10 while it's shown (only)
Click or key the "root" menu button again, flyout collapses (flyout bindings gone)

• Dominos own buttons (DominosActionButtonN buttons)

Click still works exactly as above (no issues toggling the flyout)
Hitting the keybind for the root menu button does not: Only works to show the flyout menu, but hitting the bind again does not collapse it.
No Lua errors (including from the secure system)

Tests done with the Summon Demon flyout menu on warlock.
Tests also done with ActionButtonUseKeyDown cvar on or off, behavior does not change.

Could this maybe become a feature of Dominos, in essence pulling the functionality in?

I actually considered it when I developed my own flyout implementation for retail Dominos. My preference is to get Gello's version working.

I've not replicated this with action bars 1, 2, or 14 on the cata ptr server. Tried with a simple binding of Q.