Editing the HeaderTemplate as you suggested has not worked for us yet.
When we create a copy of the Template using Blend, the template does not contain anything other than the primary control, and visual states, thus we are unable to actually override the pieces of the template.
We aren't sure which template contains the pin,close and docking options... It's the Header Grey bar with the black triangle that we are trying to hide.
We have the following choices for templates that contain headers: Bottom, Top, Left, Right, and DocumentHost.
We did try setting the VisualState and that doesn’t seem to do anything either.
VisualStateManager.GoToState(LeftPaneControlGroup, "PaneHeaderHidden", false);