Nice! I like the way you are thinking. We tend to recommend folks use Discourse (or Discourse for Teams) as the long term memory, while chat can be fine for day to day contact and getting things done. So you are on the right track.
If you already have a Discourse community, you may want to explore simply adding some private categories for internal discussion and sharing of documentation. This will be easier and cheaper for you than spinning up a new Discourse for Teams instance which will be separate.
I don’t know any examples of people using the Discourse API to share discussions between them, though I suppose it isn’t impossible. Again, you might be better off having your team just doing the manual work of retaining information that needs to be kept for the long term in Discourse. You also will want to work out as a team what goes in Discourse and what goes in chat from the get go. For example, we put documentation in Discourse, as well as any todos and longer term business-related discussions that can’t be resolved immediately.
fibery.io looks interesting - I have not seen it before. I’d be interested to hear more about it and how you are using it with Discourse and your other apps.