I agree also . . . not only does Carrot know, he uses the knowledge for the good of the watch and the city. Carrot (I think) has real respect for Vimes and respect for the way the city is run and he can see his place in the scheme of things very clearly, his role as Captain allows him the freedom to communicate (and influence) both the city dwellers and the city ruler(s). Carrot Likes his job and (again only my opinion) can see no reason to aspire to higher office when he has influence over said office anyway.
As has been stated many times in the books — no-one really knows whats going on behind Carrots innocent smile anyhow...

Regards

R.