It turns out I had the sense of the GetLocalBounds bool wrong everywhere, so ...

views/controls/button/menu_button.cc

 // Copyright (c) 2010 The Chromium Authors. All rights reserved.
 // Use of this source code is governed by a BSD-style license that can be
 // found in the LICENSE file.
 
 #include "views/controls/button/menu_button.h"
 
 #include "base/utf_string_conversions.h"
 #include "grit/app_strings.h"
 #include "grit/app_resources.h"
 #include "ui/base/dragdrop/drag_drop_types.h"
@@ skipping 108 matching lines @@
 }
 
 bool MenuButton::Activate() {
   SetState(BS_PUSHED);
   // We need to synchronously paint here because subsequently we enter a
   // menu modal loop which will stop this window from updating and
   // receiving the paint message that should be spawned by SetState until
   // after the menu closes.
   PaintNow();
   if (menu_delegate_) {
-    gfx::Rect lb = GetContentsBounds();
+    gfx::Rect lb = GetLocalBounds();
 
     // The position of the menu depends on whether or not the locale is
     // right-to-left.
     gfx::Point menu_position(lb.right(), lb.bottom());
     if (base::i18n::IsRTL())
       menu_position.set_x(lb.x());
 
     View::ConvertPointToScreen(this, &menu_position);
     if (base::i18n::IsRTL())
       menu_position.Offset(-menu_offset_.x(), menu_offset_.y());
@@ skipping 126 matching lines @@
 
 AccessibilityTypes::State MenuButton::GetAccessibleState() {
   return AccessibilityTypes::STATE_HASPOPUP;
 }
 
 std::string MenuButton::GetClassName() const {
   return kViewClassName;
 }
 
 }  // namespace views