Density-based solver is suitable for high number Mach flows (supersonic or hypersonic). If this is not the case, use pressure-based solver.
In regards to convergence issues, I would suggest to check mesh quality and start with a simpler problem without radiation. Just apply some kind of temperature boundary condition to see what is going on. The reason of divergence could be in different parts of the model and it is interesting to start with simpler cases.