this one is about twice as powerful as a typical cheap bathroom fan. It would have to go in the boxing-in you mention, because it is bigger, which will need a hatch for access to install and for maintenance.
You had better find out the actual duct size before you buy it. If you find it is 5 or 6 inches you can have a bigger fan.
Instead of a humidstat, get this one which can be wired to come on when they turn on the light, and adjust the timer for a 20 minute run-on. You say it is an internal bathroom so I guess no windows so the occupiers will probably turn on the light every time they use the room, so you are not dependent on them making a conscious effort.
It is very quiet and economical on electricity.
https://www.tlc-direct.co.uk/Products/SLTD160T.html
I haven't seen your duct. It is quite likely 100mm (four inches) but might be 125mm or 150mm. Bigger is better.
If you suck more air out, where will the air to replace it come from?
And where will the air going under the doors, through holes in the fabric, from the loft, and through trickle vents in windows be coming from?
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