Alternative: access the WebContentsView directly without an interface method.

content/browser/web_contents/web_contents_view_mac.mm

 // Copyright (c) 2012 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.
 
 #import <Carbon/Carbon.h>
 
 #import "content/browser/web_contents/web_contents_view_mac.h"
 
 #include <string>
 
@@ skipping 394 matching lines @@
     NSPoint location = [NSEvent mouseLocation];
     if ([theEvent type] == NSMouseMoved)
       webContents->GetDelegate()->ContentsMouseEvent(
           webContents, gfx::Point(location.x, location.y), true);
     if ([theEvent type] == NSMouseExited)
       webContents->GetDelegate()->ContentsMouseEvent(
           webContents, gfx::Point(location.x, location.y), false);
   }
 }
 
+- (void)setMouseDownCanMoveWindow:(BOOL)canMove {
+  mouseDownCanMoveWindow_ = canMove;
+}
+
 - (BOOL)mouseDownCanMoveWindow {
   // This is needed to prevent mouseDowns from moving the window
   // around.  The default implementation returns YES only for opaque
   // views.  WebContentsViewCocoa does not draw itself in any way, but
   // its subviews do paint their entire frames.  Returning NO here
   // saves us the effort of overriding this method in every possible
   // subview.
-  return NO;
+  return mouseDownCanMoveWindow_;
 }
 
 - (void)pasteboard:(NSPasteboard*)sender provideDataForType:(NSString*)type {
   [dragSource_ lazyWriteToPasteboard:sender
                              forType:type];
 }
 
 - (void)startDragWithDropData:(const WebDropData&)dropData
             dragOperationMask:(NSDragOperation)operationMask
                         image:(NSImage*)image
@@ skipping 92 matching lines @@
       [[[notification userInfo] objectForKey:kSelectionDirection]
         unsignedIntegerValue];
   if (direction == NSDirectSelection)
     return;
 
   [self webContents]->
       FocusThroughTabTraversal(direction == NSSelectingPrevious);
 }
 
 @end