Argh... Problem solved.
Due to a copy-and-paste bug, I was actually only setting medium and long text, not short.
It appears that if you do not set *all three texts* SAP will take the liberty of using the DDIC text whenever it feels like it, not just where you would expect. For example, in the ALV Change Layout dialog (which should be using long texts) SAP was displaying a strange variant of the DDIC text. Once I fixed my bug so I was setting short text as well as long and medium, the Change Layout dialog magically started displaying my long text for each column.
So, if you're having troubles with CL_SALV_TABLE column header text, check to make sure you are setting all three texts (short, medium and long), as setting just one or two will cause odd behavior.