I think only some rampant and unfounded speculation can help us to solve this question.
Just as Gwaihir said in the thread Hookbill linked to, the Silmaril could have traveled via the convective force within the mantle of Middle-earth and resurfaced thousands of miles away. The fact that the Lonely Mountain was, well,
lonely, suggests a volcanic origin.
Now, how can the Arkenstone be the Silmaril if it doesn't meet its appearance and size? During its travels through the mantle, it would be possible that other material has settled around it, making increasing its size. Nothing would speak against the Dwarves cutting this outer material, fashioning the resulting Arkenstone, without ever touching the Silmaril inside. The Silmaril's beauty and attraction would be diminished, but still great. Its power to burn unclean flesh could be affected, too.
It is also entirely possible that a few people, such as Gandalf, knew about its secret, but wisely concealed it so that no (more) trouble about it would emerge.