  We present MGLenS, a large series of modified gravity lensing simulations
tailored for cosmic shear data analyses and forecasts in which cosmological and
modified gravity parameters are varied simultaneously. Based on the FORGE and
BRIDGE $N$-body simulation suites presented in companion papers, we construct
500,000 deg$^2$ of mock Stage-IV lensing data, sampling a pair of 4-dimensional
volumes designed for the training of emulators. We validate the accuracy of
MGLenS with inference analyses based on the lensing power spectrum exploiting
our implementation of $f(R)$ and nDGP theoretical predictions within the
cosmoSIS cosmological inference package. A Fisher analysis reveals that the
vast majority of the constraining power from such a survey comes from the
highest redshift galaxies alone. We further find from a full likelihood
sampling that cosmic shear can achieve 95% CL constraints on the modified
gravity parameters of log$_{10}\left[ f_{R_0}\right] < -5.24$ and
log$_{10}\left[ H_0 r_c\right] > -0.05$, after marginalising over intrinsic
alignments of galaxies and including scales up to $\ell=5000$. Such a survey
setup could in fact detect with more than $3\sigma$ confidence $f(R)$ values
larger than $3 \times 10^{-6}$ and $H_0 r_c$ smaller than 1.0. Scale cuts at
$\ell=3000$ reduce the degeneracy breaking between $S_8$ and the modified
gravity parameters, while photometric redshift uncertainty seem to play a
subdominant role in our error budget. We finally explore the consequences of
analysing data with the wrong gravity model, and report the catastrophic biases
for a number of possible scenarios. The Stage-IV MGLenS simulations, the FORGE
and BRIDGE emulators and the cosmoSIS interface modules will be made publicly
available upon journal acceptance.
