Answer: The barberries are very hardy shrubs with strong constitutions so I would expect it to recover without any particular special care. These plants respond well to pruning, so the broken branch area should fill in on its own; by the end of the summer you may not even be able to tell it happened.
Although this plant can be started from rooting tip cuttings, the piece would need to be very fresh and would be taken a bit later in the season for best results.
One thing I might mention is that this plant really loves sun. If you find that it is losing some of its red coloring, that could be an indicator that it would rather be in a sunnier location. If this happens, you could transplant in very early spring next year.
Enjoy your barberry!
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