  The core-EP and BT inverses for rectangular matrices were studied recently in
the literature. The main aim of this paper is to unify both concepts by means
of a new kind of generalized inverse called $W$-weighted $q$-BT inverse. We
analyze its existence and uniqueness by considering an adequate matrix system.
Basic properties and some interesting characterizations are proved for this new
weighted generalized inverse. Also, we give a canonical form of the
$W$-weighted $q$-BT inverse by means of the weighted core-EP decomposition.
