How do I change SQL Server 2012 Availability Group DNS subdomain?

Question

Questions: how can I get my availability group to be registered as ag-ewgtest.sql.company.com given that my server thinks of itself as server.company.com.

Setup: my SQL Server 2012 availability group is on multiple subnets, similar to what's described at in this MSDN blog article. When I go into Failover Cluster Manager, it shows that name resolution is not yet available:

I then right-click "Name: ag-ewgtest" and choose properties. I check off "Publish PTR records" and it shows me that the full name is ag-ewgtest.company.com.

The problem I have is that I'm only being allowed to update the DNS zone sql.company.com. If I change the DNS name to be ag-ewgtest.sql.company.com then the full name becomes ag-ewgtest.sql.company.com.company.com.

I've tried updating my NIC as:

But hasn't made a difference. How can I change the subdomain that my availability group tries to register itself in?