This may solve my problem elsewhere...
The eve in question has a vented soffit. Can I just drop the duct "in there somewhere"? It's impossible to get to the eve from the loft and not particularly easy from the outside either as its a very narrow passage on the side of the house. This eve/soffit is around 30cm from where the bath extractor is.
Interested in your thoughts.
That's very close, I'd think you could easily push a cut length of rigid duct that far, with an elbow or something to go through a hole in the soffit, and slightly sloping so any condensation will drip outside. Though such a short length, well insulated, will not be prone to condensation as it will quickly warm. You may need to add a backdraught shutter.
You could cut your hatch and get your arm through.
The steam must not blow into the loft.