There doesn't appear to be any Barretts in Edgbaston prior to these dates,but in next door St.Martin's there is large contingent including these:
John Barrett/Ch.18/4/1752/Fa.Thomas/Ma.Mary
Siblings:
Mary/Ch.22/10/1748/Fa.Thomas
Thomas/Ch.31/8/1750/D.1750?/Fa.Thomas
Joseph/Ch.14/9/1754
William/Ch.4/7/1761
Edward/Ch.9/8/1763/D.20/8/1764
William Henry/Ch.12/11/1765
No way of verifying if this is your family but it's a probability.
Regards Jim