It copies to the clipboard by default. So if you only want to paste (i.e. no edit) you only need to do ctrl+v
But I assume you already have your hand on the mouse has you have to click and drag to make the snip, so you can then quickly click the notification as it opens, to get to the snipping tool so you can edit, and you need the mouse for that.
What is annoyingly though is if you have many screens, I have 3, then you sometimes need to travel across screens to reach the notification.