After a bit of experimentation I couldn't get it anywhere near touching the seatpost, but I could rub parts of the saddle against the rails.

On one side only, towards the front, it is possible to get enough movement in the saddle to recreate the squeak. I slipped a bit of paper between the rail and the saddle and the noise stopped. Might need to lubricate or pad this area out and see how that goes.

Nuts.