  Binary neutron star (BNS) post-merger gravitational-wave emission can occur
in the aftermath of a BNS merger -- provided the system avoids prompt collapse
to a black hole -- as a quasistable hypermassive remnant experiences
quadrupolar oscillations and non-axisymmetric deformations. The post-merger
gravitational-wave spectrum possesses a characteristic peak frequency that has
been shown to be dependent on the binary chirp mass and the neutron star
equation of state (EoS), rendering post-merger gravitational waves a powerful
tool for constraining neutron star composition. Unfortunately, the BNS
post-merger signal is emitted at high ($\gtrsim 1.5$ kHz) frequencies, where
ground-based gravitational wave detectors suffer from reduced sensitivity. It
is therefore unlikely that post-merger signals will be detected with sufficient
signal-to-noise ratio (SNR) until the advent of next-generation detectors.
However, by employing empirical relations derived from numerical relativity
simulations, we can combine information across an ensemble of BNS mergers,
allowing us to obtain EoS constraints with many low-SNR signals. We present a
hierarchical Bayesian method for deriving constraints on $R_{1.6}$, the radius
of a 1.6$\mathrm{M_{\odot}}$ neutron star, through an ensemble analysis of
binary neutron star mergers. We apply this method to simulations of the next
two LIGO-Virgo-KAGRA observing runs, O4 and O5, as well as an extended 4-year
run at A+ sensitivity, demonstrating the potential of our approach to yield EoS
information from the post-merger signal with current-generation detectors. The
A+ 4-year scenario is predicted to improve the constraint on $R_{1.6}$ from the
currently available multimessenger-based 95\% credible interval (C.I.)
uncertainty of $R_{1.6}=12.07^{+0.98}_{-0.77}$ km to
$R_{1.6}=11.91^{+0.80}_{-0.56}$ km, a 22% reduction of the 95% C.I. width.
