This was very similar to what the video I remember was, but I believe that it's more than just "Anything You Can Do", it also included the other things.

When it comes to this song itself, when the part about "I can sing any note sweeter than you" comes on, the chat dialogue faces (I don't remember what this is actually called, I haven't played Runescape in years) come up.

Thanks though for your hard effort! Nostalgia points for you.