It is always best to plant before a rain. It is possible for thatch to be too thick for light to enter, but I doubt you would have a problem. If the thatch is so thick that light can't get through, seeds certainly won't, so if you mow before you broadcast you would not have seed/soil contact. I would wait until you have rain in the forecast, Surface broadcast your radish into your standing WR and then mow it. If you have a cultipacker, I'd use that as well but it is not absolutely necessary.
Keep in mind that when WR dies, it loses much of it's moisture content. Once mowed and it makes contact with the soil, microbes speed decomposition over standing rye.
Thanks,
Jack