One month of dormancy should be fine. I would just cut the flower stem back. It is marginal how much more energy you can transfer.

Probably the reason your begonias kept growing, could be the day length you are giving them under the lights. At 12 hours or more growth will continue, get below this and the plants top growth will slow and put the energy into the tuber.Reducing day length more will make them think about going to rest. Should conditions be right, (day length and temperature)growth can continue indefinately.
You have probably cut the foliage down by now, and the tubers should start off without any problem, you could put them onto warmth and they should be growing again in 3 to 4 weeks.
The growth of tuberous begonias like many plants is governed by day length, one of the reasons why growing tuberous begonias in an area nearer the equator is almost impossible.