No, you can't: an IPv4-only device that receives that packet would interpret your extra address bytes as part of the TCP/UDP header. There simply isn't any room in the IPv4 header to squeeze extra bytes in. That is exactly why NAT baked knowledge of TCP & UDP into routers, breaking the layering design in the process. If there were any way to add extra headers to IPv4 without relying on middle boxes to support it, it would have been done a long time ago.
Not true. There is Options field to extend IPv4 header. And only server and the router closest to user would need to support such extension, keeping layering violations to minimum.
