Seems that if you want to use a custom menu bar component, there is no way to set the height and it won’t pick it up from the LookAndFeel.
Unless I’m being an idiot, the only way to get around it (short of altering the source code) is to call setMenuBar() with a dummy MenuBarModel before calling setMenuBarComponent(). This way you can either supply a height, or it will pick it up from the LnF.