Is there a 4 button (3 or 4 speed+off) fan controller that will work with Smartthings?

Devices & Integrations
2

Nothing exactly like that, but you can install any of the fan controllers that do work with smartthings from the following FAQ:

FAQ: 2020 Ceiling Fan and Dimmer light wall mount control compatible with SmartThings and Alexa?

And then add a battery operated multi button scene controller with the presets that you wanted.

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There are a number of these in different sizes and shapes.

It’s not ideal, but it might work for you.