Strictly speaking, addons don't insert their tab code into a template. They use the $extraTabs array to insert their tab information which is then used by the navigation template to build the tab. If you edit that template you will see three $extraTabs sections.
If you want to change or remove the nav tab for an addon then you need to modify the PHP code in that addon that uses $extraTabs. It would be an event listener for the "navigation_tabs" event.
edit - you can also use CSS to style or remove a tab (display: none).
Then you can edit the navigation template to add your menu.
You could also create your own listener for the "navigation_tabs" event to manipulate $extraTabs and add your menu that way. But the template edit is probably easier. The only reason to create your own listener is if you need tab selection and other functionality beyond just the dropdown menu.