  In this work we focus on a novel completion of the well-known Brans-Dicke
theory that introduces an interaction between the dark energy and dark matter
sectors, known as complete Brans-Dicke (CBD) theory. We obtain viable
cosmological accelerating solutions that fit Supernovae observations with great
precision without any scalar potential $V(\phi)$. We use these solutions to
explore the impact of the CBD theory on the large scale structure by studying
the dynamics of its linear perturbations. We observe a growing behavior of the
lensing potential $\Phi_{+}$ at late-times, while the growth rate is actually
suppressed relatively to $\Lambda$CDM, which allows the CBD theory to provide a
competitive fit to current RSD measurements of $f\sigma_{8}$. However, we also
observe that the theory exhibits a pathological change of sign in the effective
gravitational constant concerning the perturbations on sub-horizon scales that
could pose a challenge to its validity.
