I found a site that had some information about this Greek Orthodox church, it mentioned “The church, located on Balat Mahkemealtı Street, is registered in Tryphon’s list of 1583, and Paterakis’ list of 1604. In the list prepared by F. Du Cange in the second half of the 17th century it is registered as 'Theotokos Baigle.' At the end of the 18th century S. Hovhannesyan wrote that the church was mentioned as “Balinos”. The present day church was built in 1843. The church has external dimensions measuring 24.86 x 12.74 m and is about 7.8 m high. On the upper part of the iconostasis are scenes from the life of Jesus. On the lower part, from the left depictions of Hagia Trias (1845), Hagios Athanasios, Panagia Koimesis, the Virgin Mary and Child Jesus, Jesus, Ioannes Prodromos, and Hagias Menas (1846) can be seen. Moreover, on the southern wall of narthex depictions of Konstantinos and Helene (1846), and on the northern wall depiction of Hagios Panteleemon (1888) can be seen. On the surface of the ambo facing the middle nave Jesus and the four writers of the Gospels are depicted.“
The Strolling through Istanbul guidebook has slightly different dates, and considers the church "interesting only because of its great age". It has the name as Panaghia Balinu. Panaghia would imply the church is dedicated to Mary, mother of God.