  To effectively exploit large-scale structure surveys, we depend on accurate
and reliable predictions of non-linear cosmological structure formation. Tools
for efficient and comprehensive computational modelling are therefore essential
to perform cosmological parameter inference analyses. We present the public
software package ReACT, demonstrating its capability for the fast and accurate
calculation of non-linear power spectra from non-standard physics. We showcase
ReACT through a series of analyses on the DGP and $f(R)$ gravity models,
adopting LSST-like cosmic shear power spectra. Accurate non-linear modelling
with ReACT has the potential to more than double LSST's constraining power on
the $f(R)$ parameter, in contrast to an analysis that is limited to the
quasi-linear regime. We find that ReACT is sufficiently robust for the
inference of consistent constraints on theories beyond $\Lambda$CDM for current
and ongoing surveys. With further improvement, particularly in terms of the
accuracy of the non-linear $\Lambda$CDM power spectrum, ReACT can, in
principle, meet the accuracy requirements for future surveys such as Euclid and
LSST.
