  We investigate the accuracy requirements for field-level inference of cluster
and void masses using data from galaxy surveys. We introduce a two-step
framework that takes advantage of the fact that cluster masses are determined
by flows on larger scales than the clusters themselves. First, we determine the
integration accuracy required to perform field-level inference of cosmic
initial conditions on these large scales, by fitting to late-time galaxy counts
using the Bayesian Origin Reconstruction from Galaxies (BORG) algorithm. A
20-step COLA integrator is able to accurately describe the density field
surrounding the most massive clusters in the Local Super-Volume
($<135\,h^{-1}\,\mathrm{Mpc}$), but does not by itself lead to converged virial
mass estimates. Therefore we carry out `posterior resimulations', using full
$N$-body dynamics while sampling from the inferred initial conditions, and
thereby obtain estimates of masses for nearby massive clusters. We show that
these are in broad agreement with existing estimates, and find that mass
functions in the Local Super-Volume are compatible with $\Lambda$CDM.
