Connecting to VPN blocks other connections

Your client is doing this to ensure that their network is secure while you're connected to it. You should take their security as seriously as they do. If this is having a material impact on your ability to perform work for them, then speak to their security team or to the person there that you report to.

If this is preventing you from accessing sites and services unrelated to the work you perform for them, then don't access those sites and services while you're connected to the VPN.

If you attempt to circumvent their security, and their network is breached as a result, you could find yourself legally and financially responsible, which is not a position I would put myself in.