The two best attested languages for cuneiform writing are Sumerian and Akkadian. There was a time when these written languages could be equated with certain cultural groups. But, by the time of Ea-nāṣir (see more here) and King Hammurabi, they had become mostly isolated to political and linguistic categories.

For example, rulers called themselves "king of Sumer and Akkad," recognizing that the region was once historically heterogeneous. Scribes also wrote bilingual literature, using both Sumerian and Akkadian. But we are hard-pressed to find anyone calling themselves one or the other.

For rulers living in northern Mesopotamia, Akkadian were folks living around southern Iraq. For example, when King Ilu-shuma of Assur called for an andurārum, which was legislative acts focused on debt remission or economic stimulus. He established one for "the Akkadians and their dependents."

One of the ways Assyriologists have marked out cultural heritage is through naming conventions (see an example here). The name Ea-nāṣir makes sense in Akkadian, so we presume that he was culturally Akkadian. Yet, Hammurabi is a name that makes sense in Amorite and we think of him as Akkadian, too.

Moreover, Sumerian names were common even though there was no longer a Sumerian state. Not only that, but those Sumerians had Akkadian-named ancestors! Where we land today is that naming conventions provide cultural recognition, but cannot be used to secure any one specific identification.

Sumerian as a language was recorded until the end of cuneiform use (around 100 CE), but we think that the use of it as a commonly spoken language died out as early as the 2000s BCE. Later, Akkadian seems to have lost its status as lingua franca to Aramaic early in the 1st millennium BCE.

A famous example of this last point is a letter from Assyrian king Sargon II to Sin-iddina: “[You wrote], ‘...if it is acceptable to the king let me write and send my messages to the king in Aramaic.’ – why would you not write and send messages in Akkadian? Really... this is a fixed regulation!”

So, an actual response to the question: is the material associated with our copper merchant written in Sumerian or Akkadian? It's both: Akkadian, but both. More on that for another time.