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Within the next several years, pulsar-timing array programs will likely usher in the next era of gravitational-wave astronomy through the detection of a stochastic background of nanohertz-frequency gravitational waves, originating from a cosmological population of inspiraling supermassive binary black holes. While the source positions will likely be isotropic to a good approximation, the gravitational-wave angular power distribution will be anisotropic, with the most massive and/or nearby binaries producing signals that may resound above the background. We study such a realistic angular power distribution, developing fast and accurate sky-mapping strategies to localize pixels and extended regions of excess power while simultaneously modeling the background signal from the less massive and more distant ensemble. We find that power anisotropy will be challenging to discriminate from isotropy for realistic gravitational-wave skies, requiring SNR $>10$ in order to favor anisotropy with $10:1$ posterior odds in our case study. Amongst our techniques, modeling the population signal with multiple point sources in addition to an isotropic background provides the most physically-motivated and easily interpreted maps, while spherical-harmonic modeling of the square-root power distribution, $P(\hatΩ)^{1/2}$, performs best in discriminating from overall isotropy. Our techniques are modular and easily incorporated into existing pulsar-timing array analysis pipelines.