I have been pondering this enigma for many years. Recently I came across a copy of Eric Partridge's Concise Dictionary of Slang. Under the definition of "visit Lady Perriam" , slang for a visit to the lavatory, was the explanation that an underground lavatory at Oxford U. stood ( in the 1930's) on the site of a building donated by Lady Perriam.
It is my opinion that the lyricist, for the sake of alliteration and to convey a sense of mock Tudor sensibility, changed Lady Perriam to Saint Perriam.