I have found a baptism certificate in a Protestant church in Belfast, St. Ann's, for a John Brown b. 1816 (just the right time & place), son of William and Rose Brown. So far I have not been able to find much information on this family, although there is a Rose Ann Brown living a few blocks away from the church on Little York Street near where Little Patrick intersects on a valuation from the 1860's.
My next step will be to contact the church to see if they have any more information, but I thought I would check in here to see if anyone is familiar with 19th century dockside Belfast (Town parks, Shankill), or had other ideas or resources for this research.