Initially we had some problems, as we had simply scheduled a Backup Tlan to run every hour, which resulted in a couple of errors regarding hidden files that dropbox seems to generate for its syncing process and which stalled the CCC backup.
Thankfully the support for CCC is excellent and we ended up getting a solution directly from Mike Bombich, which you also might find useful.
To get it working properly you need to stop and start Dropbox via pre / post flight shell scripts, which can be attached to the Backup Task inside CCC’s > Backup Task Scheduler > Before & After panel.