While many observed ultraluminous X-ray sources (ULXs; LX>=1039 ergs s-1) could be extragalactic X-ray binaries (XRBs) emitting close to the Eddington limit, the highest luminosity ULXs (LX>3×1039 ergs s-1) exceed the isotropic Eddington luminosity for even high-stellar-mass-accreting black hole XRBs. It has been suggested that these highest luminosity ULXs may contain accreting intermediate-mass black hole (IMBH) binaries. We consider this hypothesis for dense, young (~100 Myr) stellar clusters where we assume that a 50-500 Msolar central IMBH has formed through runaway growth of a massive star. We develop numerical simulations of such clusters' cores by combining single and binary star evolutionary syntheses with a simple treatment of dynamical interactions. We model interactions of the IMBH with single and binary stars, as well as single-binary and binary-binary interactions, but not the formation of a cusp around the IMBH. The core density and velocity dispersion are assumed to be constant over 100 Myr. We investigate the succession of IMBH binary companions and the evolution of their orbital parameters to obtain estimates of the incidence of mass transfer phases and possible ULX activity involving the IMBH. We find that although it is common for the central black hole to acquire binary companions, there is a very low probability that these interacting binaries will become observable ULX sources.