I overslept and came to a completely different solution with the same visual outcome. It simply doesn’t use additional div in the li elements and p elements in the a element. The jsfiddle shows the solution thanks to the Keith J. Grant’s idea I encountered. So instead of WordPress I honed a bit my CSS skills.
But actually, the question remains: how to insert an arbitrary markup into a certain existing menu (in my case the primary menu of the chosen theme). That is into each menu item li and then into each link a, preserving the menu’s URLs.
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