  We use a combination of simulated cosmological probes and astrophysical tests
of the stability of the fine-structure constant $\alpha$, as expected from the
forthcoming European Extremely Large Telescope (E-ELT), to constrain the class
of string-inspired runaway dilaton models of Damour, Piazza and Veneziano. We
consider three different scenarios for the dark sector couplings in the model
and discuss the observational differences between them. We improve previously
existing analyses investigating in detail the degeneracies between the
parameters ruling the coupling of the dilaton field to the other components of
the universe, and studying how the constraints on these parameters change for
different fiducial cosmologies. We find that if the couplings are small (e.g.,
$\alpha_b=\alpha_V\sim0$) these degeneracies strongly affect the constraining
power of future data, while if they are sufficiently large (e.g.,
$\alpha_b\gtrsim10^{-5}-\alpha_V\gtrsim0.05$, as in agreement with current
constraints) the degeneracies can be partially broken. We show that E-ELT will
be able to probe some of this additional parameter space.
