There is increasing evidence that globular clusters retain sizeable black hole populations at present day. This is supported by dynamical simulations of cluster evolution, which have unveiled the spatial distribution and mass spectrum of black holes in clusters across cosmic age. However, black hole populations of young, high metallicity clusters remain unconstrained. Black holes hosted by these clusters mass segregate early in their evolutionary history, forming central subsystems of hundreds to thousands of black holes. We argue that after supernova feedback has subsided (greater than or similar to 50 Myr), the host cluster can accumulate gas from its dense surroundings, from which the black hole subsystem accretes at highly enhanced rates. The collective accretion luminosity can be substantial and provides a novel observational constraint for young massive clusters. We test this hypothesis by performing 3D hydrodynamic simulations where we embed discretized potentials, representing our black holes, within the potential of a massive cluster. This system moves supersonically with respect to a gaseous medium from which it accretes. We study the accretion of this black hole subsystem for different subsystem populations and determine the integrated accretion luminosity of the black hole subsystem. We apply our results to the young massive clusters of the Antennae Galaxies and find that a typical subsystem accretion luminosity should be in excess of approximate to 10(40) erg s(-1). We argue that no strong candidates of this luminous signal have been observed and constrain the subsystem population of a typical cluster in the Antennae Galaxies to less than or similar to 10-2 x 10(2) 10 M (circle dot) black holes, given that feedback does not significantly impede accretion and that the gas remains optically thin.