I'm afraid you're going to need some extra hardware, so the easiest solution would actually be to get a real Classic Controller. Read on for the way of connecting the Switch controller:
There's a thread on here about using the Pro Controller on Wii:
https://gbatemp.net/threads/switch-procon-on-wii.584290/
The My Mate Vince video on there shows you how to do it with a Brook PS4 -> Wii U Super Converter (which is now marketed as being for Switch - in both cases you would need to change the firmware to the Classic one, which make the controller act as a Wii Classic Controller) and a Mayflash Magic-NS to make the controller appear to the converter as a DS4.
Alternatively you could get an 8bitdo Retro Receiver to plug into a Wiimote, but as it's meant for the SNES Classic, it doesn't recognise all the buttons.