Replicate SQL Data Inside Firewall to Outside Firewall - Suggestions
The method I'm familiar with would be to to only have an internal database. The firewall separating your internal and DMZ networks can be configured to only allow appropriate communications through, in your case 1443 (or your SQL port) between the web server and database.  

However, you can't simply rely on your firewall for security. Proper SQL permissions would be necessary to only allow access to your web data. And of course you should be validating user input to protect against SQL injection. 

If you do want to go the replication route, check out the MS offerings such as SQL Server Transactional Replication.