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.

What class and flyout menu if I can ask? So I can try on PTR as well to have parity.

Ok, tried to do a bit more thorough testing and I think I figured it out. It's on the FlyoutBindings side moreso than Dominos.
Your Q binding was a nice clue 👍🏻

Addons enabled: Dominos/Dominos_Config, FlyoutBindings, BugGrabber + BugSack
Testing with the Warlock Summon Demon flyout again, dragged to the 2nd Dominos bar.

That flyout for a L85 warlock has 6 items (6 pet types)
My particular use case was essentially creating a double-tap summon.
6:root menu, (dynamic binds) 6:felguard, 5:felhunter, 4:Imp, 3:Voidwalker .. etc

This allows me to 6-6 to summon felguard, 6-5 felhunter, 6-4 imp .. you get the idea.

I also tried with not overlapping root, so 7:root, 6:pet1,5:pet2 etc.

In both cases I observed the behavior I reported initially.

It turns out this is in no way a Dominos issue. FlyoutBindings binds 1-10 on menu open regardless how many items are there (so even if 6 items, all 1-10 are temporarily bound)
Me binding the root menu to 7 (even with 6 menu items) didn't work because once the menu opens 7 is now bound to a non-existent menu entry, not the root button.
It is not intuitive when you can observe 6 items in the menu and root is bound to 7 (or 8 or 9) but it's logical.

Sorry for wasting your time.
PS. I went back and tested default actionbars (no Dominos) as well as Dominos Bar1 that reuses them and the behavior is consistent, the problem is FlyoutBindings "reserving" all 1-10 binds when the menu opens so the root button cannot be bound to a number.

PPS. It also means that while not intuitive (in the sense that you cannot hit 7 again to cancel out of the menu as you could do with mouse clicks, I can still use them to double-tap summon with the caveat that to cancel out the menu I have to actually start a summon and interrupt it (not fat-fingers friendly)