It's exactly that - blossom end rot (BER). With courgettes, it is often associated with the female blossoms not getting pollinated.
Try pollinating by hand. I use a small artist's paint brush to gather pollen from male blossoms (long thin stem, internal structure a single fuzzy column covered with pollen; pollen can also be gathered where it drops from the column into the flower).
Transfer the pollen with the brush onto the reproductive part of female flowers. Female flowers have a shorter stem, shaped like a miniature fruit. The interior reproductive structure is complex, looks sort of brain-like. That structure is where you wan to deposit the pollen.
I did this all summer and have produced many courgettes. It is best done in the morning while the flowers are open and fresh, each flower is open for only a part of the day, usually morning.
Good luck!