الْبَكْرِيِّ: the Bakrite, genitive by the preceding construct relation.
بِشْرٍ: the person’s name, a genitive explanatory apposition.
The بِ of «بِشْرٍ» is part of the name, not a preposition. The first noun is genitive through iḍāfah; the second follows its case as an apposition.