The new LA4533 chip finally showed up.
Firstly, it is in a slightly different package to the Sony CX20089A. It is slightly smaller, so I had to bend a couple of pins slightly to make it fit.
Secondly, it's mute function is different. On the Sony chip, the mute function is active high, and on the LA4533, it is active low. A PNP transistor pulls the mute pin high for a second when you first turn the Walkman on to prevent any click noise. To make this feature work with the LA4533, I simply replaced said PNP transistor with an NPN.
I'm pleased to say the Walkman now works
.