I'd just like to add to the list of Asana and Team Drive users hoping for integration soon. No pressure. Well maybe a little.
For everyone else in this thread, a workaround we've been doing is to attach a file while it's still in My Drive, and only move it to a Team Drive afterwards. The link doesn't change in the move so it doesn't invalidate the attachment from within Asana. It's more inconvenient for files that are already in a Team Drive. An admin has to move it out, attach it, then put it back. When we're lazy we sometimes end up just adding a link in the comments, but then that won't show up in the Files section.
That being said, it seems that it's just the file picker that needs to be updated to display Team Drives since the files themselves aren't off-limits. I don't want to assume that that's easy for the Asana team to do or even if Google already allows 3rd parties to do it, but I really hope the implementation isn't too far off.
Hope this helps.