I’ve written a small plugin to enable sharing posts on LinkedIn, just like the other sharing buttons.
You can find the code here: GitHub - rcfox/Discourse-LinkedIn-Sharing-Button: Add a button in Discourse to share posts on LinkedIn
You can see it in action on the Upverter Community forum.
There seems to be a slight bug in Discourse around this though. The order in the “share links” setting doesn’t seem to be respected for the button added by the plugin. It’s probably coming in late…