Brick wall

[Maiden Stone]

I couldn't see a baptism in Mayo to fit your Bridget. She may have been from a parish which doesn't have baptism registers in those years.
Existing Belmullet baptism registers begin 1841.
https://registers.nli.ie/parishes/0742
Existing baptism registers in parishes adjacent to Belmullet start later. Some parishes have marriage registers from earlier years than baptism registers. Click on a parish name on the little map on Belmullet page for information about registers in other parishes. Then click the "more information" link at foot of each parish page for a list of existing registers. 

[Maiden Stone]

I have no idea where in Ireland except Mayo, I also have a second cousin looking he thinks it might be Bellmullet

Do you know why your cousin thinks Bridget might have been from Belmullet? Is there any family lore passed down?