The answer was given by Rayleigh in his Theory of Sound, which you can read on The Internet Archive. Here's the relevant part:
The relevant solution to the differential equation (1) in Rayleigh for the tangential component has the \(cos(m\theta)/m\) behavior.
But, I wasn't able to start with Rayleigh's coordinates and end up with his equation (1).
Here's my work; where did I go wrong?