Do you have the Dock at the bottom?
And are you moving the cursor to the very bottom of the external screen?
Do you have the Dock automatically hiding when not in use?
You could try turning Dock automatic hiding on and seeing if that helps.
System Settings->Desktop & Dock, turn on "Automatically hide and show the Dock"
then move the cursor against the bottom of each screen - does the Dock now appear on each display as you move there?
On a sidenote: what is it that you need the Dock so much for? As someone who does not use the Dock at all, I am always somewhat curious as to what other people use it for, and why it is so important. I can work a whole day and never touch, or see, the Dock.