How to select item in Chrome's toolbar overflow menu

I’m trying to select a particular menu item in Chrome’s toolbar “overflow” (three dots) menu. But this isn’t working out for me - I can’t figure out how to address the menu.

Here’s the script I have so far.

tell application "Google Chrome" to activate

tell application "System Events"
	tell application process "Google Chrome"
		tell window 1
			tell toolbar 1
				
				tell (first button whose accessibility description is "Chrome")
					click
					
					(* How to select my menu item now? *)
					
					
				end tell
			end tell
		end tell
		
	end tell
end tell

Anyone know how to go from here?

Do I understand correctly, these 3 points?

Which menu point, etc. do you want to select?

I got exactly to the same point as you did. It seems this button is similar to that button in Acrobat.

The only way I found to select a menu item is with keystrokes, for example to select the Cast… menu item:

tell application "Google Chrome" to activate
tell application "System Events"
  tell application process "Google Chrome"
    tell window 1
      tell toolbar 1
        tell button "Button"
          perform action "AXPress"
        end tell
      end tell
    end tell
  end tell
  keystroke "c"
  keystroke return
end tell

However you can assign shortcuts to the menu items of this button via System Preferences > Keyboard > Shortcuts. But I guess you know that.

Cast is exactly what I’m going for. Keyboard shortcuts FTW! Thanks!

Since I do not know with scripts, I would have given the tip with the shortcut also @iNik.
So that you can easily forget the shortcut use a palette with BetterTouch Tool. Check out this video here. Use the pallets in all my apps.