I've got a quick workaround. I just added a handler for the Physics Controller's mouse move event as so:
pc = cnvGame.GetValue(PhysicsControllerMain.PhysicsControllerProperty) as PhysicsControllerMain;
pc.MouseMove += new PhysicsControllerMain.MouseMoveHandler(touchMove_MainMenu);
So while the mouse (touch) is 'moving', the line can be found and it's visibility overridden:
void touchMove_MainMenu(string sprite)
foreach (UIElement uiElement in cnvGame.Children)
if (uiElement.GetType().Name == "Line")
uiElement.Visibility = System.Windows.Visibility.Collapsed;
Note that this only works for me because I've got not other Line UIElements in the game canvas. If you do utilize other Lines, you'd have to rule these out to find the physic's controller's one.
Also, I realize that going through the entire UIElementCollection to find this particular line is probably the least efficient way to do it, but I've got no other ideas at the moment.