  Joint lensing and dynamical mass profile determinations of galaxy clusters
are an excellent tool to constrain modification of gravity at cosmological
scales. However, search for tiny departures from General Relativity calls for
an accurate control of the systematics affecting the method. In this analysis
we concentrate on the systematics in the reconstruction of mass profiles from
the dynamics of cluster member galaxies, while assuming that lensing provides
unbiased mass profile reconstructions. In particular, in the case study of
linear $f(R)$ gravity, we aim at veryfying whether in realistic simulations of
cluster formation a spurious detection of departure from GR can be detected due
to violation of the main assumptions (e.g. dynamical equilibrium and spherical
symmetry) on which the method is based. We aim at identifying and calibrating
the impact of those systematics by analyzing a set of Dark Matter halos taken
from $\Lambda$CDM N-body cosmological simulations performed with the GADGET-3
code. [...] If no selection criteria are applied, $\sim 60\%$ of clusters in a
$\Lambda$CDM Universe (where GR is assumed) produce a spurious detection of
modified gravity. We find that the probability of finding cluster in agreement
with GR predictions $P_{GR}$ mainly depends on the properties of the halo's
projected phase-space and on shape orientation of the cluster along the
line-of-sight projection. We define two observational criteria which correlate
with the probability to find clusters in agreement with GR predictions and
which can be used to select [...] those objects that are more suitable for the
application of the proposed method. In particular, we find that according to
these criteria the percentage of spurious detection can be lowered down to
$\sim 20\%$ in the best case. Our results are relevant in view of data that
will be available with the next generation surveys.
