Stop Lightroom Classic from downloading images from Lightroom Cloud

Do note that if you are using Lightroom Cloudy, it will still download copies to your local hard disk for speed of access reasons. It will dynamically manage the local storage and offload images you haven't used in a while but they still need to be local to work on them. The only way, as far as I know to work with raw images just on the cloud is to use https://lightroom.adobe.com. This runs Lightroom cloudy in a web browser. The functionality is more limited even than desktop Lightroom cloudy but it does work. 

In the end what you are asking for is a version of Lightroom Cloudy that has the features of Lightroom Classic. That is quite far off still unfortunately. The pace of adding features that we need to Cloudy has not been fast and in the mean time, Classic has been gaining features. They are almost as far apart as they were when the cloud Lightroom first came out.

"There is an app for that." It's called Lightroom (for the cloud). If you are subscribed to the photography plan then it's available for you to download. If your main goal is to have images in the cloud then that is the version you want to be using. The feature set is a little different, and it seems to be designed for a different type of user. But if your main goal is to have your images in the cloud then that is what you should be using.

Not possible. Lr-Classic must have a 'local' copy to allow editing.

If you set Lightroom-Classic to Sync then it will automatically download all files (those imported to 'Mobile' devices) from the Cloud ecosystem.

There is possibly an 'idea' thread for this function at the Feedback forum (I cannot locate)- Adobe Photoshop Family -Lr Classic