It sounds like you are describing an efficient process already that takes a minute to do initially and a minute once the document is ready and to be checked in again. Use Edit Document to check it out and copy the checked out file to Teams. Then copy it back to the check out location in Docman once done in Teams. Considering the value created by the efficient collaboration in Teams, those two minutes of overhead does not seem like a high price to pay.
Two macros in Docman can probably automate most of this. Once the file is checked out, the macro could copy it to some location where Teams, or at least SharePoint/OneDrive can pick it up. Then a check in macro could do the reverse.
Most probably more advanced automation can be done as well, leveraging the APIs available in Teams/SharePoint. Sounds like a fun project to work on :)
Good luck!
PS. We are thinking about ways on how to integrate Docman into MS Teams.
Thanks @Mathias Dahl for the answer. With IFS23R1 released at this moment, I think I heard in one of the webinars on new functionality, that a Teams integration is available. Teams, and IFS, are rather big as applications, so I was wondering if you know anything about this?
I don't know much about that I'm afraid. I recommend that you refer to the release material that should be available now, or soon (not sure where though...)