It is not clear from your post where the VPN client software is running (on the host or VM?) Assuming the VPN client is running on the host, that probably explains why the VM needs to switch to NAT as the bridged settings of the VM would have nothing set for VPN access while the NAT would route through the host VPN.
As for access to Google (or any site outside of company intranet), you have to ask the VPN/IT admins whether access outside of the company intranet are routed by the VPN. Some company VPNs are set up exclusively for intranet or only partially to public internet for certain sites. This might be due to IT security reasons or for cost/bandwidth reasons (it does not want to route an employee/contractor streaming movies) or for both. Anyway, the loss of access to Google while on VPN may not have anything to do with VMware Workstation.
There is nothing stopping you from adding another virtual NIC that has bridged while the other has NAT. But that brings along routing issues as well with a multi-homed machine. Even if you get multi-NICs to work, you might be violating IT policy of the company. It is best to ask your VPN/IT admins. It is probably not worth getting a reprimand or something worse if you are found to be violating company IT policy (having a machine whether physical or virtual that can serve as a bridge for traffic from possible harmful public internet sites into company intranet).
