hi ceejay, just had a wee look at your post, first i would ask if your are sure her name is bethea? because the scottish census transcribed her name as such on the 1851 census does not mean it is correct.
next i would say that on the 1861 census in penrith where the family is now, your bethea would be 14, a good chance now that she could be a servant close by.
i reckon you will need to get the birth cert first and then we can trace her through the census.
wishing you well and trying to help
Joe