We consider the monomial expansion of the $q$-Whittaker polynomials
given by the fermionic formula and via the {\em inv} and {\em quinv}
statistics. We construct bijections between the parametrizing sets of
these three models which preserve the $x$- and $q$-weights, and which
are compatible with natural projection and branching maps. We apply
this to the limit construction of local Weyl modules and obtain a new
character formula for the basic representation of
$\widehat{\mathfrak{sl}_n}$.