I'm not sure what scenario you are trying to implement. It seems you start the browser and get an authentication popup when you open some URL using MWG as the proxy. You type your credentials in the browser and start surfing, you are no longer required to type your credentials again since your browser cached them in memory. At some point you want your browser to forget the user name/password you've entered, but you don't want to restart the browser, correct? I would say that restarting the browser is the only reliable way to do it.
You can setup the proxy in a way that it tells your browser that authentication failed, forcing the browser to present you an authentication popup again but this solution won't work in all cases. Some plugins in your browser, e.g. FoxyProxy may continue to use last known good credentials. The browser still will be able to access some pages over persistent connections if you are using NTLM to authenticate. Better restart the browser.