We present a detailed numerical investigation of the tunability of a diffusive random laser when Mie resonances are excited. We solve a multimode diffusion model and calculate multiple light scattering in presence of optical gain that includes dispersion in both scattering and gain, without any assumptions about the parameter. This allows us to investigate a realistic photonic glass made of latex spheres and rhodamine and to quantify both the lasing wavelength tunability range and the lasing threshold. Beyond what is expected by diffusive monochromatic models, the highest threshold is found when the competition between the lasing modes is strongest and not when the lasing wavelength is furthest from the maximum of the gain curve.