  Modelling nonlinear structure formation is essential for current and
forthcoming cosmic shear experiments. We combine the halo model reaction
formalism, implemented in the REACT code, with the COSMOPOWER machine learning
emulation platform, to develop and publicly release REACTEMU-FR, a fast and
accurate nonlinear matter power spectrum emulator for $f(R)$ gravity with
massive neutrinos. Coupled with the state-of-the-art baryon feedback emulator
BCEMU, we use REACTEMU-FR to produce Markov Chain Monte Carlo forecasts for a
cosmic shear experiment with typical Stage IV specifications. We find that the
inclusion of highly nonlinear scales (multipoles between $1500\leq \ell \leq
5000$) only mildly improves constraints on most standard cosmological
parameters (less than a factor of 2). In particular, the necessary modelling of
baryonic physics effectively damps most constraining power on the sum of the
neutrino masses and modified gravity at $\ell \gtrsim 1500$. Using an
approximate baryonic physics model produces mildly improved constraints on
cosmological parameters which remain unbiased at the $1\sigma$-level, but
significantly biases constraints on baryonic parameters at the $>
2\sigma$-level.
