The OED calls rinktum "rare southern U.S. dialect" and "alteration of RECTUM", with a quotation from Faulkner in 1929 ("I'll skin your rinktum").

Webster's 3rd had not heard of that, but does list "rinktum ditty: [origin unknown]: a mixture of tomato sauce, onion, cheese, egg, and seasonings served on toast".

I suspect that the use in choruses is mock Latin like "harum scarum", that "rinktum ditty" was arbitrarily named by someone who knew one of those choruses (an inmate of a summer camp would be a good guess), and that any allusion to the other end of the digestive system was secondary.