the Ballaugh line is covered in www.manxnotebook.com/famhist/families/acaine.htm

all entries there are from land deeds (actual documents for which I have copies), marriage contracts or entries from the parish registers. This is the family that descended from Patrick the twin son of Adam - it is only in the deed settling the estate on their son Patrick did I find his mother's family name. Patrick was still around for many years in Ballaugh post your sightings of a Patrick Cain in America.

The Kk Michael Adam Caine husband of Elinor Stephan family, with which I first confused the children, is at www.manxnotebook.com/gazateer/ml_oryd.htm - again the Patk Cain of the age to match your sightings in America is noted by land records etc as being on the Island

The Ann Cain als Corlett is from a completely unrelated Cain family - the reasoning behind this I have given ad nuseam - Kk Michael had many Cain families whose children shared relatively few first names - some of these who resided in the Bishop's demesne are particularly difficult to research as parish registers for Michael + Ballaugh have significant gaps and these families left no land records.

I cannot know of any of your family traditions re Manx descent, personally I think you are pinning too much on the lost bible but without any record of the entry it cannot at the moment help - it may be that there is a Manx descent but not from the Patrick Cain you suppose, however the Cain name is not entirely Manx eg in your area of the USA in the 1881 census there were many Cain families among the decendants of slave families - there are a very few Manx in America at this time, some went c. 1715 possibly due to Jacobite sympathies making life at home uncomfortable, some may have gone for trade (eg Henry Callister in 1740's went as an apprentice to Liverpool + was sent onto Maryland), somewhat later post 1765 some emigrated for ecconomic reasons. So far, and I have probably looked at over 60% of all 18th Century Manx wills (and all Michael + Ballaugh wills) there is no mention of any Cain or other Manx in this area whereas all other manx emigrants are known by such records since if they had any right to property putative heirs needed to convince a court that they were dead otherwise the court would write into any decree a clause reserving their rights.