Cosmological information is usually extracted from the Lyman-$\alpha$ forest correlations using only either large-scale information interpreted through linear theory or using small-scale information interpreted by means of expensive hydrodynamical simulations. A complete cosmological interpretation of the 3D correlations at all measurable scales is challenged by the need of more realistic models including the complex growth of non-linear small scales that can only be studied within large hydrodynamical simulations. Past work were often limited by the trade off between the simulated cosmological volume and the resolution of the low-density intergalactic medium from which the Lyman-$\alpha$ signal originates. We conduct a suite of hydrodynamical simulations of the intergalactic medium, including one of the largest Lyman-$\alpha$ simulations ever performed in terms of volume (640 $h^{-1}\mathrm{Mpc}$), alongside simulations in smaller volumes with resolutions up to 25 $h^{-1}\mathrm{kpc}$. We compare the 3D Lyman-$\alpha$ power spectra predicted by those simulations to different non-linear models. The inferred Lyman-$\alpha$ bias and RSD parameters, $b_\alpha$ and $\beta_\alpha$ are in remarkable agreement with those measured in SDSS and DESI data. We find that, contrary to intuition, the convergence of large-scale modes of the 3D Lyman-$\alpha$ power spectra, which determines $\beta_\alpha$, is primarily influenced by the resolution of the simulation box through mode coupling, rather than the box size itself. Finally, we study the BAO signal encoded in the 3D Lyman-$\alpha$ power spectra. For the first time with a hydrodynamical simulation, we clearly detect the BAO signal, however we only marginally detect its damping, associated with the non-linear growth of the structures.