Since Step it out Mary is a modern song written by Sean McCarthy (see thread Step It Out, Mary), Irish approaches to the spelling of the name (now or in the past) would seem irrelevant.

McCarthy said specifically that he based the chorus on a children's skipping song:

Step it out Mary, my fine daughter
Step it out Mary, if you can
Step it out Mary, my fine daughter
Cock your legs for the country man.

Probably our correspondent Mary's father had just tacked a form of that onto the end of another song that he only knew in part; that sort of thing happens all the time. The thing to be looking at would be "I'm going to marry that little Irish girl", I'd guess. I'd expect that to be a music hall or vaudeville piece, likely American, but perhaps we'll get better information soon.

Probably the discussion would be best kept in the one thread rather than spread over several; and, with the best will in the world, it would be considerate to type in the normal way rather that all in UPPER CASE, which is hard on the eye and has a tendency to annoy people.