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

chrome/browser/ui/views/extensions/extension_view.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_VIEW_H_
 #define CHROME_BROWSER_UI_VIEWS_EXTENSIONS_EXTENSION_VIEW_H_
 #pragma once
 
 #include "build/build_config.h"
 
@@ skipping 10 matching lines @@
 class ExtensionView : public views::NativeViewHost {
  public:
   ExtensionView(ExtensionHost* host, Browser* browser);
   virtual ~ExtensionView();
 
   // A class that represents the container that this view is in.
   // (bottom shelf, side bar, etc.)
   class Container {
    public:
     virtual ~Container() {}
-    // Mouse event notifications from the view. (useful for hover UI).
-    virtual void OnExtensionMouseMove(ExtensionView* view) = 0;
-    virtual void OnExtensionMouseLeave(ExtensionView* view) = 0;
     virtual void OnExtensionPreferredSizeChanged(ExtensionView* view) {}
   };
 
   ExtensionHost* host() const { return host_; }
   Browser* browser() const { return browser_; }
   const Extension* extension() const;
   RenderViewHost* render_view_host() const;
   void DidStopLoading();
   void SetIsClipped(bool is_clipped);
 
   // Notification from ExtensionHost.
   void UpdatePreferredSize(const gfx::Size& new_size);
-  void HandleMouseMove();
-  void HandleMouseLeave();
 
   // Method for the ExtensionHost to notify us when the RenderViewHost has a
   // connection.
   void RenderViewCreated();
 
   // Set a custom background for the view. The background will be tiled.
   void SetBackground(const SkBitmap& background);
 
   // Sets the container for this view.
   void SetContainer(Container* container) { container_ = container; }
@@ skipping 45 matching lines @@
   // Note: the view does not own its container.
   Container* container_;
 
   // Whether this extension view is clipped.
   bool is_clipped_;
 
   DISALLOW_COPY_AND_ASSIGN(ExtensionView);
 };
 
 #endif  // CHROME_BROWSER_UI_VIEWS_EXTENSIONS_EXTENSION_VIEW_H_