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The anomalously fast orbital decay of the black hole X-ray binaries A0620−00 and XTE J1118+480 has been attributed to dynamical friction in dark matter density spikes, and annihilation within these spikes has recently been proposed as the origin of the high-energy positron excess measured by AMS-02. That proposal treats positron propagation using interstellar conditions throughout, neglecting the environment close to the black hole. In this work, we use equipartition estimates of the accretion-flow magnetic field, calibrated to these two systems, to test whether that region can safely be neglected. We show that it cannot: synchrotron losses in the near-horizon field allow only a small fraction of a percent of the positrons to escape, weakening the predicted flux by more than three orders of magnitude. We further show that the spike is limited by its own annihilation, suppressing the source independently of the magnetic field. Either effect alone is sufficient to remove the proposed explanation. We conclude by discussing what the same environment implies for the use of these binaries as dark matter targets.