For fast T1-weighted imaging, zero echo time (ZTE) imaging provides rapid sampling of 3D k-space, but lacks T1 contrast due to its small flip angle excitation with short hard pulses. To efficiently increase T1 contrast, longer phase-modulated pulses can be used. However, longer pulses lead to larger dead-time gaps and pulse profile-weighted images. Here, we formulate an inverse problem to directly reconstruct profile-compensated images from multi-coil data without any intermediate step for dead-time infilling. We demonstrate that leveraging coil sensitivities and alternating phase-modulated excitations sufficiently condition the inverse problem, allowing for iterative reconstructions of T1-weighted acquisitions.