He was considered a Syrian Rav because he served the community for many years.
He was also accepted by them as such, despite being from an Iraqi background, because he was considered a tremendous Torah scholar and was highly respected among his fellow Chachamim including Rav Ovadia Yosef (his chavrusa) and Rav Ben Zion Abba Shaul with whom he studied together in the illustrious Yeshivat Porat Yosef. The Syrian community was very smart to not let his non Syrian background to stand in the way of seeing his greatness.
Furthermore he was considered one of the foremost talmidim of Rav Ezra Attias who was himself was born in Aleppo, Syria and that carried a lot of weight and credibility within the Syrian community, therefore he was not considered an outsider.
Moreover, Rav Ben Chaim was married to the daughter of Rav Yaakov Kassin. Rav Yaakov was the chief rabbi of the Syrian community and that also gave him legitimacy within the community.