You can format your special category template so that it is recognized as a page template by how the initial comment is written.Pages#Creating_Your_Own_Page_Templates You would then use the Rewrite API to get your page loaded instead of the usual category template any time a /category/ permalink is used.
In the end, you end up with the same thing, except perhaps it's slightly easier to edit any static content, so I'm not sure I see the point. But then, I don't have to, it's not my site :)
I should understand your reference to menus though, but I don't. What menu points to front end pages? The back end menu points to a back end category page which alters the categories stored in the database. This is not directly related to how categories are displayed on the front end.