No, there's no way to do what you're looking for. The browser cache option should only be used on sites where you do not perform authentication. If you do authentication, there's no way for the browser to know that the page being loaded is different from the cached copy.
The System - Cache plugin is disabled by default for just this reason.. it can cause issues in many different scenarios. In addition to that, the Browser Caching option is also disabled by default. Browser caching is only recommended for very select sites where content doesn't change often and never on a per-user basis.
Just to note, you'd have the same results with the standard Joomla Login module.
I hope that helps explain,
Alex