Please don't get hung up on spelling of names.
Until the late 19th century there was no such thing as 'correct' spelling. Names, generally, were spelled the way the clerk writing them down thought they should be written, and some clerks had wonderfully inventive ideas about it. This might or might not be how the people concerned would have spelled them, assuming that they could read and write (though I imagine that your Cochran(e)s were educated enough to be able to do so).
My own great-grandfather spelled his name in half a dozen different ways when he registered the births of his children, and that was in the 1870s and 1880s.
I have been told that there are 53 different recorded spellings of the surname Taylor, and there are other names with at least as many. I have a family in my tree who spell their surname Story, Storey, Storry, Storrie, Storie and Storrey without any apparent rhyme or reason for changing from one version to another until the various branches fixed their own versions in the lat 19th century or early 20th century.
So the registration of a name as 'Cochran' instead of 'Cochrane' is not a 'mistake', it's just a variant spelling. You will almost certainly waste a lot of time and effort if you try to read anything significant into it.
