Electron injection constitutes a central unresolved problem for diffusive shock acceleration. Here we study perpendicular nonrelativistic collisionless shocks in a regime of high Mach numbers, as appropriate for young supernova remnants. We use high-resolution large-scale two-dimensional fully kinetic particle-in-cell simulations that sample a representative portion of the turbulent shock front and account for time-dependent effects of cyclic shock reformation. The microphysics of perpendicular shocks in weakly magnetized plasmas is governed by ion reflection from the shock that leads to the formation of magnetic filaments in the shock ramp resulting from ion-beam filamentation instabilities, and also electrostatic Buneman modes in the shock foot. The latter can provide electron injection through shock-surfing acceleration. Recent studies show that additional electron acceleration can also occur due to spontaneous magnetic reconnection triggered within magnetic filaments in the turbulent shock transition. We study conditions allowing for efficient electron pre-acceleration by this process. We demonstrate a dependence of the magnetic-reconnection rate on the temperature of the inflowing plasma and also on numerical parameters of the simulations, such as the ion-to-electron mass ratio and the configuration of the average magnetic field with respect to the simulation plane. We discuss the resulting electron spectra and the relevance of our results to the physics of fully three-dimensional systems.