I am not sure how you would use that switch you linked.
It is just 2/3/4 Position 8 Terminals Rotary Cam Changeover Switch with Screws. Insulated to 660V, able to handle 20A reliably.
And how is the fan wired? Is it "3-speeds motor" ( like a ceiling fan), or is what is in that little box.... probably just a 3600rpm 2pole motor, or 1800rpm 4pole motor connection( typically common).... ?
Off the wall voltage, you have a cycles per second. When a (AC) motor is driven to speed ( designed) it uses this (cycles per) to commute its rpm based on the pole/armature structure and a few other things..
You cannnot change this, Hz from the AC on the wall... ( in the USA here it is 60Hz, IDK what it is where your at... 50Hz? )...
Andyway.... the only way to rally change the RPM of a wall fan is with a frequency converter.. A Variable frequency drive (VFD). They do exist and are expensive in industry but cheap at the scrap yard.
It is that, or change the pitch of the blade, or restrict its (in)flow... but that is flow mechanics.. and they get.. er uhm.. complicated. You know. Complicated, as in it might be simpler to buy a typical HVAC duct fan.. right rpm and flow.
Duct it and choke it. Running in vacume means very little power is used.. and very little air is moved... Or VFD it. Or go to Home Depot and buy a proper duct air mover. They are gentle and made to be unobtrusive and/or not create great turbulence.
Maybe an old rheostat dimmer ( wastes the extra current as heat) would work.. , but I am pretty sure a solid state dimmer would not work, based on how they do their thing. And how the fan is probably wired for its three speeds.. nothing like a control system or anything.... . Modern dimmers are built from semiconductors instead of variable resistors, because they have higher efficiency. A variable resistor (Rheostat dimmer) would dissipate power as heat and acts as a voltage divider.
Variable frequency drives are mainly used to drive induction motors at variable speed.
The speed of an induction motor is proportional to the frequency of AC supply. Simple,Change frequency to get required speed.
Torque of induction motor is proportional to square of voltage. Auto transformers can only change voltage that changes torque which is undesirable.
VFD is not only used for controlling the speed of an induction motor, although it is the main function of any VFD.
There are lots of other important benefits of using a vfd than a dimmer, autotransformer etc.
It protects your motor from over-running under over current condition (overcurrent limit can be set in parameters of a vfd).
Acceleration & De-acceleration of a motor during startup and stopping of a motor can be controlled as per requirement using a VFD. Which is an important requirement for a lot of applications.
Speed vs. Voltage profile can be controlled using a VFD which is very important in different voltage ranges such as 400V, 380V, 440V or 480V operations. Proper setting does improve alot the current consumption of a motor.
Power saving is also a big advantage of using a VFD as it uses PWM(Pulse Width Modulation) technique to deliver the output voltage signal.
there are many other advantages of using a VFD to run a motor.
Typical Rheostat Fan Dimmer... if you are not wasting any real power nor have access to VDF systems... dont mind waste.
Pay attention to the V, I, and Hz, for the one you buy: Rheostats are sized this way.
https://www.amazon.com/dp/B015SLCJ2...glD1hSnmcEKu8mVRbKopBH6T4HZ4ZZOZqC_qInTE6a3ao
