The thrust of your comment is more or less correct, but you make several significant errors in the details.
Akkadian, not Sumerian was the language of international diplomacy in the period you speak of. Sennacherib came later in the Iron Age. And if the Egyptians needed cedar, they'd write to Byblos (which during the New Kingdom, was a tributary state more often than not), not Assyria.
The original comment is almost certainly thinking of https://www.amazon.com/dp/0199858683/ (Brotherhood of Kings: How International Relations Shaped the Ancient Near East).
It's a very good book.
A minor theme is indeed that Mesopotamian city-kings had a hard time trading with Egypt. Mesopotamia seems to have produced mostly textiles, which Egypt made domestically. What Egypt needed from international trade was copper and wood.
All the Mesopotamian city-kings wanted gold, though, which came from Egypt. They could get this one time, by marrying a daughter to the Pharaoh. We have an example of Pharaonic correspondence with a king who repeatedly asked for unworked gold, finally sent a daughter over, received a mountain of gold in return, and concluded by complaining that the gold had been worked (into statues).
I assume many of those statues got melted down.
There is some discussion in https://www.amazon.com/dp/0670022667/ (Carthage Must Be Destroyed: The Rise and Fall of an Ancient Civilization) of how the Phoenician city-states of the bronze age (source of cedar wood to Egypt) gradually succumbed over the centuries to Assyrian empire-building. It's not the main focus of the book though.
Akkadian, not Sumerian was the language of international diplomacy in the period you speak of. Sennacherib came later in the Iron Age. And if the Egyptians needed cedar, they'd write to Byblos (which during the New Kingdom, was a tributary state more often than not), not Assyria.