The geolocation API which allows websites to obtain your location from your web browser is supposed to be opt-in. The standard specification says that it must not be enabled without the users permission. For that reason it can be switched on and off in the settings of most web browsers.
When you don't trust your web browser to respect that preference, then you should use a different one. When you run software you can not trust, you've already lost any security you could have.
That being said, you could build a setup like most corporations do. Set up a LAN where the workstations have only a local IP address and no internet connection. Then set up a proxy server on your LAN which is the only machine with both a LAN IP and a WAN IP. Install the VPN client on the proxy so it uses the WAN IP only to tunnel to the VPN provider. Have the web browsers on the workstations connect to that proxy using the proxys LAN IP address. That way you could do web browsing while your browsing machine isn't aware of the WAN IP (and not of the VPN provider either, for that matter). As a side-effect you have also succesfully firewalled your workstation against any internet communication which isn't supported by the proxy.