Replicating contained SQL databases to customer in near real-time
Interesting problem to solve. I don't have a lot of familiarity with contained databases but after some quick research I understand their general purpose and the challenges you're facing.
The few ideas that come to mind are possibly looking into Log Shipping or Database Mirroring. You might be able to schedule more frequent data synchronizations with one of these methodologies than routinely sending a BACPAC file.
Finally, it looks like that Contained Databases do support Availability Groups but I'm not sure if that'll be of much help since AlwaysOn Availability Groups require a Windows Failover Cluster to be setup between the servers of the Availability Group, which I'm assuming is not conducive to your business scenario.

After some more brainstorming, research, and re-reading what you wrote, there may be a way to leverage a cloud provider like Azure, as you mentioned. This DBA.StackExchange answer gives a few options for moving data into an Azure instance. Additionally it reminded me of SSIS which you should be able to leverage to synchronize data from your instance within your DMZ to an external server (whether that's in Azure or on-prem).