As far as I know, the Hittite and the Assyrian were Bronze Age empires but one of the reasons why they conquered other people was because of their mass produced iron weapons. Mycenaean used bronze but I believe they expanded their influence based on trading, not fighting.
So, apart from the Egyptians, did any civilization of the Bronze Age became an Empire without developing iron weapons and tools?
This is probably only a partial answer, but when the Hyksos invaded Egypt and established the 15th dynasty they didn't have iron weapons, relying instead on the chariot and composite bow.
In ancient Mesopotamia there are other examples of empires that didn't use iron weapons, such as the Akkadian Empire that started in the 24th century BCE. Another one is the Old Assyrian Empire (2025-1750 BCE), in which, as the Wikipedia article informs, iron was known but was so rare that it was more valuable than gold. A third one in Mesopotamia is the Babylonian Empire (Hammurabi's reign seems to have gone from 1792 to 1750 BCE, although these dates are disputed by some sources).
I have used some Wikipedia articles for this answer, and a lot of information about these peoples can be found in Mario Liverani's The Ancient Near East.
