Hello Caro5961, there are church records for 5 James McGivern's in Irish Church records on Roots Ireland all born in County Down, 4 RC and one C of I between 1810 and 1820, none in 1815, this may be that church records for this period do not exist for other parishes so no guarantee any of them is correct without parents names, the 4 RC are born in the one parish Kilbroney which is on the north side of Carlingford Lough in Northern Ireland now. The other person is much further north in the county in the parish of Magheralin.
We rely on church records for the pre 1864 period of civil BMDs and 1845 for C of I records, these church records are incomplete prior to this for various reasons so you have reached a brick wall with out knowing the parents names even if records exist.
There is a County Down site herehttps://rosdavies.com/ and there is an active County Down private Facebook page dealing with DNA. I live about 30 miles south of the County Down border and the McGivern name is still about.
Good Luck
Pat