The radio goes into the charger base much easier once you realize that the radio (actually the battery pack) has little grooves and the charger base has little guide rails that slot into the grooves. You have to align the grooves with the guide rails.
Pay attention to this little unmentioned detail and it becomes much easier.
By the way, a battery pack alone (all by itself) can also be charged in the same charger base. The magic grooves make it possible.