There have been suggestions to check the Gilbert & Sullivan societies, and I found at least two lists of songs Sullivan wrote. It wasn’t a huge list, but I doubt whether many have been commercially recorded, which is possibly why it does not turn up on databases:
I think the combined expertise of these societies will probably produce the answer, particularly if it was written by Sullivan
Of course it is possible that it wasn’t written by Sullivan, and he was just playing a music hall song of the period. The musical director of the film will have the answer. Someone had the expertise to choose this song for its dramatic effect.
It is a great film, and I have a minor interest in Gilbert & Sullivan and their works. I get obsessive about finding the answers to obscure questions, and so I understand your point of view.