This behavior is as it should be.
If you use 2 Exchange-Servers and 1 File Share Witness only the site that has the majority can run the DAG.
By design you're not able to run the 2nd Database in the 2nd Site while the 1st Site is running.
DAG is Windows Cluster and if you have both sites up and running the same time and the internet connection everything would break.
There is one feature to run both mailboxes in one site without majority. It's called Datacenter Activation Mode.
As today this only works with 3 DAG-Servers. Starting with Exchange 2010 SP1 Microsoft said that this will be supported with two Exchange Server within a DAG. Never the less it is not supported to run one Database in one site while you're internet connection is failed.
http://technet.microsoft.com/en-us/library/dd979790.aspx
