After the last episode, the iRobot folks sent out an entirely new Roomba unit. That did the trick. The problem was not with the charger or the battery.
Now I have rather a lot of spare parts ... an extra charger and battery, the carcass of the dead Roomba (minus the wire grill sent back as proof). Add to this the two battery-powered "virtual wall" gadgets, and wall-mounting bracket -- for all of which it turned out I have no need.
Assuming the lifetime for this unit is a bit longer, I will not be needing replacement parts for quite a while.
I am tempted to take apart the dead unit to see what the inside bits involved in charging look like.