I was able to figure out a short cut:
LoginName loginName = HeadLoginView.FindControl("HeadLoginName") as LoginName;
if (loginName != null && session != null)
{
loginName.FormatString = "Full Name";
}
This finds the LoginName element in the LoginView, and then captures the value by setting the FormatString. I wish there was a more elegant way to do this. If anyone knows, send it your own way.