I've been thinking about the frame for this episode, and this is the best thing that I've come up with so far:
The method of making dragon-proof gates was developed by the Dwarves of Moria in cooperation with the Elves of Hollin and used for the gates of Moria. Oropher, who was king of the Woodland Realm at the time, commissioned the Dwarves of Moria to make dragon-proof gates for his halls. He pays the dwarves in installments as they work on the gates. The final touch on the gates is an inscription in Quenya. Oropher did not realize the inscription would be in Quenya, so he is outraged when the Dwarven craftsmen show him the completed gates and they are covered in the language of the Kinslayers. Oropher would be particularly sensitive to this because he was in Doriath when the Second Kinslaying happened and may have lost friends or family in the attack. The Dwarves cannot change the inscription to Sindarin because the magic would not work the same, so Oropher refuses to pay the final installment and throws the craftsmen out. Thranduil was elsewhere in Middle-earth when all of this happened, and Oropher did not speak of it to his son because he quickly began to appreciate the gates despite the Quenya inscription.
The document that had the Moon-letters is biased from a Dwarven perspective and only cryptically refers to what Oropher was upset about. It might say he refused to pay them because he "disliked the language used in the inscription" or something to that effect, which Thranduil would not realize referred to Quenya and Dain would see as a pointless quibble about a magic inscription that cannot be altered because he would not understand or may not even know about the ban or its implications.
The frame scenes for this episode can focus on tense negotiations as the kings trying to figure out exactly what the ancient conflict described in the document is now that the hostage situation has ended.
Beginning scene: Negotiations with Thranduil, Dain, Bard, Gandalf, and Bilbo. Dain is demanding payment. Thranduil is insisting that his father would only refuse payment if there was a just cause. Bard says let's find out what the cause is, but they do not know how to do this. Thranduil does not remember the incident because he was not there for it. Gandalf is certain the answer can be found in the document. The cryptic hint about the language makes Bilbo think back to when he saw the gates in Mirkwood. He confides in Gandalf that something about them seemed odd, but he can't put his finger on it.
Ending scene: Bilbo has a eureka moment in the middle of the negotiations. He realizes that the inscriptions were in Quenya and extrapolates the reason Oropher did not pay for them. The issue is finally understood, but Dain still demands payment, so the hostilities are not resolved.
