I want the gross / net switch to only affect the display in the shop. The plugin should have no influence on the order process. Is this possible?

Yes, this is possible. In the plugin settings under Settings > Extensions > Price Gross / Net Switch, you will find a configuration option called "Impact". Select "Restricted: Not in cart and order process". This ensures the gross/net toggle only affects the frontend display – the cart and order process remain unaffected and always use net prices.