I discovered a transcript, sorted by name, which covered Falstone, Greystead, Humshaugh, Simonburn, Thorneyburn and Wark 1813 - 1839.
There were only 7 baptisms for Batty, Batey or Baty in that time span and all those parishes. No Bridget unfortunately.
She wasn’t in Bellingham Presbyterian either.
Perhaps the parishes on the other side of Bellingham next time?
Christine