I frequently use VPN.
The VPN client should be providing you with DNS resolution. We use a
Cisco VPN client.
If the machine is known as GDIHQ internally I would still refer to it as
GDIHQ externally, once I've initiated the VPN.
I never use an IP address. During a Mimix switch the IP address will
change. I could try connecting to my internal address of 10.10.something
and it would fail because our DNS and whatnot has been updated to point to
the other machine. And, we stop that interface on the primary machine.
There are extremely rare cases when one has to stop the DNS client service
and restart it because even ipconfig /flushdns wouldn't fix it. We had
one user who had that alot. We switched out that CEO. Problem solved.