In #6.13 Comrades in Arms part 2, when Hawkeye and Margaret are demonstrating the new vascular surgery technique at the 8063rd, Margaret mentions that the clamp they are using was developed at the 4077th. That is the clamp that is developed in this episode. Margaret was referring to a clamp that supposedly hadn't been developed yet. This was due to the fact that, like many TV series, the installments were not shown in the order in which they were produced.
As Margaret is flinging scrubs around looking for her ring, one of the garments lands on Charles' head. In the next cut shot it is covering more of his head than when it first landed.
It is doubtful that the surgeons wouldn't check Mr. Shin's inscription on the ring before giving it to Margaret.
When Hawkeye and BJ are talking to Mr. Shin about the possibility of buying a ring to replace the one that Margaret lost, a plastic canteen can be seen hanging from Mr. Shin's cart. These canteens weren't in use by the Army until 1962, several years after the Korean War ended and could not have been on his cart.