it could drag in humid air from the rest of the house
It is most unlikely that the rest of the house could contain air as humid as the room where you have steamy showers and hot baths
my opinion is that a fan will have little effect if any
Why is it your opinion that removing water vapour from the bathroom will have "little effect if any?"
If the customer wants one, and the room is particularly humid, I'd suggest something more powerful than the £12 cheapies that builders like to fit.
Small 100mm fans usually have an extract rate of about 80 cu.m/hr which is hardly enough, even in a well-heated and insulated room without a shower. If you fit a 6-inch fan it can be much more powerfil (if there is not already a duct in the wall then there is no need for a little fan)
To be effective, you will get better results if it comes on with the light switch and has a timed run-on. 5 minutes may be enough to freshen the air, but after a steamy shower it will probably need 20-30 minutes because the walls, tray, and towels will need to dry off. Humidstat ones are I find nowhere near as good, though the idea often appeals.
If you can fit a ducted one above the ceiling, you can get more power and less noise.
Modern, good quality fans with ball-bearing motors can be very quiet. Noise is a thing that householders moan about, and many of them refuse to use fans if they are noisy.
This is a good-quality 6" fan with timer. There are cheaper brands that are not as good. If you make a 6" hole and the customer insists on a cheap, weedy fan, it can be replaced later with little effort.
https://www.tlc-direct.co.uk/Products/SLS300CRZ.html
If looking at others, compare the noise (db) and the airflow (cu.m/hr)
This horrible cheap fan, loved by builders, is the worst one I know. Observe that it is noiser and not as powerful.
https://www.screwfix.com/p/manrose-xf100s-15w-bathroom-extractor-fan-white-240v/11097
Good luck!