  Extending previous work on this subject, we evaluate the impact of
vector-like fermions at next-to-leading order accuracy in models with a massive
vector leptoquark embedded in the $SU(4)\times SU(3)^\prime\times SU(2)_L\times
U(1)_X$ gauge group. Vector-like fermions induce new sources of flavor symmetry
breaking, resulting in tree-level flavor-changing couplings for the leptoquark
not present in the minimal version of the model. These, in turn, lead to a
series of non-vanishing flavor-changing neutral-current amplitudes at the loop
level. We systematically analyze these effects in semileptonic, dipole and
$\Delta F=2$ operators. The impact of these corrections in $b\to s\nu\nu$ and
$b\to c\tau\nu$ observables are discussed in detail. In particular, we show
that, in the parameter region providing a good fit to the $B$-physics
anomalies, the model predicts a $10\%$ to $50\%$ enhancement of
$\mathcal{B}(B\to K^{(*)}\nu\nu)$.
