It appears there are way more variants of the Business CD stereo than I realised:

https://e46radio.com/category/e46-head-units/page/2/

Mine is apparently the Siemens VDO one, but according to that page it should be compatible with Aux.

The only other possibility I can think of is that the cables I bought don’t work because they don’t appear to have resistors on them like the OEM one does, which is apparently the thing that makes the head unit recognise the Aux input.

Anyone have any other thoughts?