We investigate the effect of k-space broadening of the interlayer pairing kernel on the critical temperature Tc and the k-dependence of the gap function in a one-dimensional version of the interlayer pair-tunneling model of high-Tc superconductivity. We consider constant as well as k-dependent intralayer pairing kernels. We find that the sensitivity to k-space broadening is larger the smaller the width of the peak of the Fermi-level gap calculated for zero broadening. This width increases with the overall magnitude of the interlayer tunneling matrix element, and decreases with the bandwidth of the single-electron intralayer excitation spectrum. The width also increases as the Fermi level is moved towards regions where the excitation spectrum flattens out. We argue that our qualitative conclusions are valid also for a two-dimensional model. This indicates that at or close to half-filling in two dimensions, when the Fermi-surface gap for zero broadening attains its peaks at (±π/a, 0) and (0, ±π/a) where the excitation spectrum is flat, these peaks should be fairly robust to moderate momentum broadening.