Lasing in the bad cavity regime has promising applications in precision metrology due to the reduced sensitivity to cavity noise originating from cavity length fluctuations. Here we investigate the spectral properties and phase behavior of pulsed lasing on the S-1(0) - P-3(1) line of Sr-88 in a mK thermal ensemble, as first described by S. A. Schaffer et al. [Phys. Rev. A 101, 013819 (2020)]. The system operates in a regime where the Doppler-broadened atomic transition linewidth is several times larger than the cavity linewidth. We find that for some detunings of the cavity resonance, the influence of the cavity noise on the peak lasing frequency can be eliminated to first order despite the system not being deep in the bad cavity regime. Experimental results are compared to a model based on a Tavis-Cummings Hamiltonian, which enables us to investigate the interplay between different thermal velocity classes as the underlying mechanism for the reduction in cavity noise. These velocity-dependent dynamics can occur in pulsed lasing and during the turn-on behavior of lasers in the superradiant crossover regime.