  We investigate the approximations needed to efficiently predict the
large-scale clustering of matter and dark matter halos in beyond-$\Lambda$CDM
scenarios. We examine the normal branch of the Dvali-Gabadadze-Porrati model,
the Hu-Sawicki $f(R)$ model, a slowly evolving dark energy, an interacting dark
energy model and massive neutrinos. For each, we test approximations for the
perturbative kernel calculations, including the omission of screening terms and
the use of perturbative kernels based on the Einstein-de Sitter universe; we
explore different infrared-resummation schemes, tracer bias models and a linear
treatment of massive neutrinos; we employ two models for redshift space
distortions, the Taruya-Nishimishi-Saito prescription and the Effective Field
Theory of Large-Scale Structure. This work further provides a preliminary
validation of the codes being considered by Euclid for the spectroscopic
clustering probe in beyond-$\Lambda$CDM scenarios. We calculate and compare the
$\chi^2$ statistic to assess the different modelling choices. This is done by
fitting the spectroscopic clustering predictions to measurements from numerical
simulations and perturbation theory-based mock data. We compare the behaviour
of this statistic in the beyond-$\Lambda$CDM cases, as a function of the
maximum scale included in the fit, to the baseline $\Lambda$CDM case. We find
that the Einstein-de Sitter approximation without screening is surprisingly
accurate for all cases when comparing to the halo clustering monopole and
quadrupole obtained from simulations. Our results suggest that the inclusion of
multiple redshift bins, higher-order multipoles, higher-order clustering
statistics (such as the bispectrum) and photometric probes such as weak
lensing, will be essential to extract information on massive neutrinos,
modified gravity and dark energy.
