I have found two baptisms at about the right time, if Alice was indeed Alice Burrel(l):
19 Sep 1748 Alice Burrell, daughter of Robert Burrell, baptised at St Hilda's, Hartlepool
5 July 1747 Alice Burrell of Panns, born 5 Apr 1747 daughter of John Burrell, at St Michael's, Bishopwearmouth
There is also the baptism at St Hilda's of Alice Burrel, daughter of William and Margaret Burrel on 30 Jul 1738, but I think this would make Alice too old as she would have been nearly 50 in 1785.