Spectral anti-crossings between the fundamental guided mode and core-wall resonances alter the dispersion in hollow-core anti-resonant-reflection photonic crystal fibers. Here we study the effect of this dispersion change on the nonlinear propagation and dynamics of ultrashort pulses. We find that it causes emission of narrow spectral peaks through a combination of four-wave mixing and dispersive wave emission. We further investigate the influence of the anti-crossings on nonlinear pulse propagation and show that their impact can be minimized by adjusting the core-wall thickness in such a way that the anti-crossings lie spectrally distant from the pump wavelength.