How to specify DNS on iOS when connected via VPN?
The DNS settings over a VPN connection should be forced by the VPN server. This makes sure DNS queries get routed securely over the VPN, and allows you to see private services on the intranet that may not be exposed as public DNS records.

It's possible the VPN server may not be telling iOS what DNS server to use, something that could be solved by a configuration change in the VPN server?