  The paper is the third of a series dedicated to the ab initio description of
monopole giant resonances in mid-mass closed- and open-shell nuclei via the
so-called projected generator coordinate method. The present focus is on the
computation of the moments $m_k$ of the monopole strength distribution, which
are used to quantify its centroid energy and dispersion. First, the capacity to
compute low-order moments via two different methods is developed and
benchmarked for the $m_1$ moment. Second, the impact of the angular momentum
projection on the centroid energy and dispersion of the monopole strength is
analysed before comparing the results to those obtained from consistent
quasi-particle random phase approximation calculations. Next, the so-called
energy weighted sum rule (EWSR) is investigated. First, the appropriate ESWR in
the center-of-mass frame is derived analytically. Second, the exhaustion of the
intrinsic EWSR is tested in order to quantify the (unwanted) local-gauge
symmetry breaking of the presently employed chiral effective field theory
($\chi$EFT) interactions. Finally, the infinite nuclear matter
incompressibility associated with the employed $\chi$EFT interactions is
extracted by extrapolating the finite-nucleus incompressibility computed from
the monopole centroid energy.
