Would you use Basecamp’s chat? Or mostly their messaging system for communication? If the latter, I can see a case of using both. But that could also lead to people not taking Basecamp communication seriously.
We use each level of communication for different reasons. (is it a quick question? Is it a well thought out idea that needs attention? Is it time sensitive?) It feels very natural and after a short period your organization falls into a groove as long as Basecamp is the ONLY method for work communication outside conference calls, etc.