How would you fully and properly speak Thorin's name, "Thorin II Oakenshield"?
Would you say "Thorin Oakenshield the Second"? Or "Thorin, the Second, Oakenshield"?
Same question goes for Dain II Ironfoot, Thorin III Stonehelm, and many others.
The second way. Thorin Oakenshield is not the second "Thorin Oakenshield". Oakenshield is a nickname which applies only to him and not to his ancestor Thorin I. So the correct way to render it is "Thorin the Second, Oakenshield."
Or if you want a long-winded version that borrows from Game of Thrones, "Thorin, second of his name, called Oakenshield."
For a real world example, Alexander the Great was King Alexander III of Macedon, but he is not known as "Alexander the Great the Third". The "great" appellation does not apply to the previous two Alexanders because they didn't conquer the known world. If you want to specify both his number and nickname, it would be "Alexander the Third, the Great".