  The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed
$\mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and
Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named
authors obtained a Mellin-Barnes integral representation for these
eigenfunctions. In the present paper, we develop the analytic theory of the
$b$-Whittaker functions from the perspective of quantum cluster algebras. We
obtain a formula for the modular open Toda system's Baxter operator as a
sequence of quantum cluster transformations, and thereby derive a new modular
$b$-analog of Givental's integral formula for the undeformed Whittaker
function. We also show that the $b$-Whittaker functions are eigenvectors of the
Dehn twist operator from quantum higher Teichm\"uller theory, and obtain
$b$-analogs of various integral identities satisfied by the undeformed
Whittaker functions, including the continuous Cauchy-Littlewood identity of
Stade and Corwin-O'Connell-Sepp\"al\"ainen-Zygouras. Using these results, we
prove the unitarity of the $b$-Whittaker transform, thereby completing the
analytic part of the proof of the conjecture of Frenkel and Ip on tensor
products of positive representations of $U_q(\mathfrak{sl}_n)$, as well as the
main step in the modular functor conjecture of Fock and Goncharov. We conclude
by explaining how the theory of $b$-Whittaker functions can be used to derive
certain hyperbolic hypergeometric integral evaluations found by Rains.
