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

views/controls/native/native_view_host.cc

 // Copyright (c) 2009 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/native/native_view_host.h"
 
 #include "base/logging.h"
 #include "ui/gfx/canvas.h"
 #include "views/controls/native/native_view_host_wrapper.h"
 #include "views/controls/native/native_view_host_views.h"
@@ skipping 103 matching lines @@
 
   if (visible) {
     // Since widgets know nothing about the View hierarchy (they are direct
     // children of the Widget that hosts our View hierarchy) they need to be
     // positioned in the coordinate system of the Widget, not the current
     // view.  Also, they should be positioned respecting the border insets
     // of the native view.
     gfx::Insets insets = GetInsets();
     gfx::Point top_left(insets.left(), insets.top());
     ConvertPointToWidget(this, &top_left);
-    gfx::Rect local_bounds = GetLocalBounds();
+    gfx::Rect local_bounds = GetContentsBounds();
     native_wrapper_->ShowWidget(top_left.x(), top_left.y(),
                                 local_bounds.width(),
                                 local_bounds.height());
   } else {
     native_wrapper_->HideWidget();
   }
 }
 
 void NativeViewHost::Paint(gfx::Canvas* canvas) {
   // Paint background if there is one. NativeViewHost needs to paint
@@ skipping 64 matching lines @@
 // NativeViewHost, private:
 
 void NativeViewHost::Detach(bool destroyed) {
   DCHECK(native_view_ || views_view_);
   native_wrapper_->NativeViewDetaching(destroyed);
   native_view_ = NULL;
   views_view_ = NULL;
 }
 
 }  // namespace views