As per my understanding , there is no permission to prevent a user from editing the `Navigation menu` for Lightning Console apps.

As stated in the [documentation][1] : 

> Admins define which default items are available in an app’s navigation
> menu, but user can change certain parts of the user interface on his/her
> own. User can add items to the item menu, and rename or remove the
> items added. User can’t rename or remove default items that your
> admin specified for the app, including any custom or standard objects.


If your admin defines basic menu items and users add their own, `Edit` button should not be a issue for you since these menu items are applied only to the user who modifies it.

  [1]: https://help.salesforce.com/s/articleView?id=sf.console_lex_personalize_nav.htm&type=5