I saw that Dorico does not yet have a feature for percussion stickings but how can I make the snare drum sound as played with brushes?

You would need to choose a drum kit patch that includes brush sounds in Play mode. See here for the basics of how to do that.

Does this mean I have to download a new sound library for the drum set? In the list I see Halion Sonic SE, Noteperformer and Play. My Play sounds don’t include brushes.

There is a brush kit in the HSSE library that you already have.