James,
No - the above config would initiate a VPN tunnel to come up, but I am sure that all unknown internet traffic would still break out from the PIX. I presume you have a default route in the PIX that either point out of the outside interface or onto the next hop device from your provider.
The danger is if you change the default route to a internal HQ layer 3 routing device ot remote end ASA - you will loss the VPN connection, as the external IP of the HQ will be routeable over the VPN, but cannot establish!
In the past I have routed internet traffic over a VPN tunnel via a dynamic routing protocol and GRE tunnels, do you have this capability?
HTH>