  [Abridged] High-resolution N-body simulations have recently shown that the
structural properties of highly nonlinear cosmic structures, as e.g. their
average concentration at a given mass, could be significantly modified in the
presence of an interaction between Dark Energy and Dark Matter. While a
constant interaction strength leads to less concentrated density profiles, a
steep growth in time of the coupling function has been shown to determine a
large increase of halo concentrations over a wide range of masses, including
the typical halos hosting luminous spiral galaxies. This determines a
substantial worsening of the "cusp-core" tension arising in the standard
$\Lambda $CDM model and provides a direct way to constrain the form of the Dark
Energy interaction. In the present paper we make use of the outcomes of some
high-resolution N-body simulations of a specific class of interacting Dark
Energy models to compare the predicted rotation curves of luminous spiral
galaxies forming in these cosmologies against real observational data. Our
results show how some specific interacting Dark Energy scenarios featuring a
steep growth in time of the coupling function -- which are virtually
indistinguishable from LCDM in the background -- cannot fit the observed
rotation curves of luminous spiral galaxies and can therefore be ruled out only
on the basis of dynamical properties of small-scale structures. Our study is a
pilot investigation of the effects of a Dark Energy interaction at small
scales, and demonstrates how the dynamical properties of visible galaxies can
in some cases provide direct constraints on the nature of Dark Energy.
