  The Universe's assumed homogeneity and isotropy is known as the cosmological
principle. It is one of the assumptions that lead to the
Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) metric and it is a cornerstone
of modern cosmology, because the metric plays a crucial role into the
determination of the cosmological observables. Thus, it is of paramount
importance to question this principle and perform observational tests that may
falsify this hypothesis. Here we explore the use of galaxy cluster counts as a
probe of a large-scale inhomogeneity, which is a novel approach for the study
of inhomogeneous models, and to determine the precision with which future
galaxy cluster surveys will be able to test the cosmological principle. We
present forecast constraints on the inhomogeneous Lema\^{\i}tre-Tolman-Bondi
(LTB) model with a cosmological constant and cold dark matter, from a
combination of simulated data according to a compilation of `Stage-IV' galaxy
surveys following a methodology that involves the use of a mass function
correction from numerical $N$-body simulations of an LTB cosmology. When
considering the \lcdm fiducial model as a baseline for constructing our mock
catalogs, we find that our combination of the forthcoming cluster surveys, will
improve the constraints on the cosmological principle parameters as well on the
FLRW parameters by about $50\%$ with respect to previous similar forecasts
performed using geometrical and linear growth of structure probes, with
$\pm20\%$ variations depending on the level of knowledge of systematic
effects.These results indicate that galaxy cluster abundances are sensitive
probes of inhomogeneity, and that next-generation galaxy cluster surveys, will
thoroughly test homogeneity at cosmological scales, tightening the constraints
on possible violations of the cosmological principle in the framework of
$\Lambda$LTB scenarios. (Abridged)
