Have ExtensionHost use TabContents instead of RenderViewHost. Try #3.

chrome/browser/ui/views/extensions/extension_popup.h

 // Copyright (c) 2011 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.
 
 #ifndef CHROME_BROWSER_UI_VIEWS_EXTENSIONS_EXTENSION_POPUP_H_
 #define CHROME_BROWSER_UI_VIEWS_EXTENSIONS_EXTENSION_POPUP_H_
 #pragma once
 
 #include "base/memory/ref_counted.h"
 #include "chrome/browser/extensions/extension_host.h"
@@ skipping 65 matching lines @@
   virtual void BubbleGotFocus(BrowserBubble* bubble);
   virtual void BubbleLostFocus(BrowserBubble* bubble,
                                bool lost_focus_to_child);
 
   // content::NotificationObserver overrides.
   virtual void Observe(int type,
                        const content::NotificationSource& source,
                        const content::NotificationDetails& details);
 
   // ExtensionView::Container overrides.
-  virtual void OnExtensionMouseMove(ExtensionView* view) { }
-  virtual void OnExtensionMouseLeave(ExtensionView* view) { }
   virtual void OnExtensionPreferredSizeChanged(ExtensionView* view);
 
   // The min/max height of popups.
   static const int kMinWidth;
   static const int kMinHeight;
   static const int kMaxWidth;
   static const int kMaxHeight;
 
  private:
   ExtensionPopup(Browser* browser,
@@ skipping 21 matching lines @@
 
   content::NotificationRegistrar registrar_;
 
   // The observer of this popup.
   Observer* observer_;
 
   DISALLOW_COPY_AND_ASSIGN(ExtensionPopup);
 };
 
 #endif  // CHROME_BROWSER_UI_VIEWS_EXTENSIONS_EXTENSION_POPUP_H_