My father, who was born in 1899 in western Kentucky, and whose forebears probably came from the Appalachians, did indeed use "rinctum" to mean "rectum." He wasn't very musical, but he had relatives who were. As far as I know, he didn't know any song that used "rinctum" as a refrain. If he had, he probably would have considered it risqué, unfit for mixed company, and certainly unfit for a children's song.

I suspect that "rectum" was felt to be a "foreign" (i.e. Latin) word used primarily by doctors—who, after all, often eschew plain language, and are fond of using "contusion" for "bruise," "axilla" for "armpit," and so on. (Plain language would be "asshole.")

My father had a peculiar bias against accepting "foreign" words into his vocabulary. I don't know whether this was an idiosyncrasy of his, or a characteristic of the community he grew up in. It was as if, on hearing an unfamiliar word, he tended to assume the speaker was mispronouncing it, and really meant something else. Then, if he needed to use the word himself, he would often change the pronunciation to something he was already familiar with.

It makes sense to me that, if my father (or someone like him) had already known the word "rinctum" from a folksong, and then heard a doctor refer to a "rectum," he would have assumed that the intended word was really "rinctum."

Something similar was probably going on when "asparagus" came to be called "sparrow grass" in some communities.

Does this help, or am I only spelling out the obvious?