It actually works very well with private browsing. You just need to have the extension enabled for private browsing (in iOS 17 that's a separate switch) and tap

It actually works very well with private browsing. You just need to have the extension enabled for private browsing (in iOS 17 that's a separate switch) and tap on "get your session link" once in the extension menu in Safari. That way it knows about your subscription, even in private tabs.