We investigate the linewidth enhancement factor of the type-II red InGaN/GaNSb/GaN quantum-well (QW) laser, employing a non-Markovian gain model with many-body effects included. It is shown that the linewidth enhancement factor of the type-II red InGaN/GaNSb/GaN QW structure is almost independent of the peak-gain coefficient. This behavior is contrasted with that of the conventional type-I InGaN/GaN QW structure, whose linewidth enhancement factor increases as the peak-gain coefficient increases. These results can be explained by the peak-gain dependencies of the differential refractive-index change and the differential gain. Moreover, the type-II red InGaN/GaNSb/GaN QW laser yields much smaller values of the linewidth enhancement factor than the conventional type-I InGaN/GaN QW laser. The type-II red InGaN/GaNSb/GaN QW laser with a relatively small, excitation-independent linewidth enhancement factor is expected to be highly useful for many practical applications.