Although the earliest bronze objects have been dated to the fourth millennium, which is considered the beginning of the Bronze Age, bronze does not become common until the Middle Bronze Age (2000-1550 BC), which is when weaponry, and then tools, began to be predominantly made from bronze. So that's why ancient Sumer is considered part of the Early Bronze Age, but actual bronze artifacts from that period are fairly rare, since most tools and weapons were still stone or copper.

Also copper is rather soft, so I guess it would not be of much use as a material for weapons.

Although copper is soft and does not hold an edge well, it is still suitable for weapons and armor, and was actually prized by Mesopotamian cultures for its martial applications. Pure copper is most suitable to maces and axes and began to be used for weapons in the Chalcolithic period, with the earliest known example from the end of the sixth millennium. Copper ore is sometimes naturally alloyed with arsenic, which increases its hardness and makes it more suitable for blades and helmets. Regardless, there are many extant examples of copper spearheads, axeheads, maceheads, arrowheads, and helmets from the Early and Middle Bronze Ages. Look up the finds from the Cemetery of Ur to see some copper weaponry from the Early Dynastic period.

Sources:

Dan Howard, Bronze Age Military Equipment

William Hamblin, Warfare in the Ancient Near East to 1600 BC