St Phillips graveyard is indeed the area around the cathedral, but was far more extensive than it is today. Several years ago it was discovered during some renovation work that there were bodies buried well outside the present day limits.
At the time you are searching, St Phillips was just another city centre parish church. It wasn´t made into a cathedral until (as far as I can remember) Birmingham became a city.
Birmingham Central Library has extensive parish records for St Phillips, but no idea about availability at the moment as they are working on restricted hours until the move to the new library is complete.