  Understanding the behavior of the matter power spectrum on non-linear scales
beyond the $\Lambda$CDM model is crucial for accurately predicting the
large-scale structure (LSS) of the Universe in non-standard cosmologies. In
this work, we present an analysis of the non-linear matter power spectrum
within the framework of interacting dark energy-dark matter cosmologies (IDE).
We employ N-body simulations and theoretical models to investigate the impact
of IDE on these non-linear scales. Beginning with N-body simulations
characterized by a fixed parameter space delineated by prior observational
research, we adeptly fit the simulated spectra with a simple parametric
function, achieving accuracy within 5\%. Subsequently, we refine a modified
halo model tailored to the IDE cosmology, exhibiting exceptional precision in
fitting the simulations down to scales of approximately 1 h/Mpc. To assess the
model's robustness, we conduct a forecast analysis for the Euclid survey,
employing our refined model. We find that the coupling parameter $\xi$ will be
constrained to $\sigma(\xi) = 0.0046$. This marks a significant improvement by
an order of magnitude compared to any other current observational tests
documented in the literature. These primary findings pave the way for a novel
preliminary approach, enabling the utilization of IDE models for observational
constraints concerning LSS data on non-linear scales.
