goes to Google, on the raw network, and on the VPN.
You can't "go" to a destination on two networks in a single request. It's all packets on a wire, if it comes from two sources, it was two requests.
Unless you mean two different requests. As in while on the VPN everything is tunneled, and while not on the VPN it's not, but this is the opposite of what the OP was asking for. He wants the VPN on for some use cases, and off for others. That's split tunneling.
He'll likely wind up with difficulties around trying to figure out which destinations he doesn't want routed through the VPN, because there's no way to do it by protocol, since routing happens on layer 3, not 4 or 7. He'll likely need to know those address in advance.
Raw disk access is a privilege in Linux, usually reserved for root.
You could have root change the permissions on the directory to allow another user or group write access.