Based on further clarification from the below comment(s), my suggestion would be to set up a DNS server at your primary location as well, configure it to use your upstream public/ISP DNS server for its resolution, and configure conditional forwarding for your secondary locations domain to use the secondary location's DNS server.
If your secondary location used
sec.example.com as it's domain, for example, all requests (
google.com, etc) would go through your primary site DNS server to the public DNS, unless the lookups were for the
sec.example.com domain. In that case, the request would be passed through to the secondary location DNS server for resolving.