In a lot of sim games, its a common trope for one civilization to be fighting another civ with superior metalworking.
Are there any large scale instances of armies clashing where it was a straight up battle of bronze vs iron or iron vs steel? Or perhaps just very different grades of the same metal? What were the results? Did the inferior weapons break the first time they clashed swords/spears? Or Maybe the difference in metals was not as important when fighting with spears and shields as it would be in a Hollywood-style fight between sword wielding fighters?
This is not directly answering your question, but rather correcting a misconception you seem to have. Bronze is not necessarily a much worse material than the rudimentary low-grade steel they would be using in the transition period. The real answer to the Bronze/Iron age divide is not that Iron is stronger, but rather, it is more abundant. What characterizes Iron Age civilizations from their Bronze Age predecessors was really more about the amount of arms and armour that could be fielded and the size of the armies that could be raised.
See, while copper, one of the primary components of Bronze, is extremely common, tin, the other major component (and the one required to turn malleable copper into hard Bronze) is not. When supplies of tin were available, copper/bronze is a far better material to work with than Iron because it is much easier to smelt and mold; Iron's melting point is 500 degrees hotter, and quite frankly, the technology required to smelt Iron is actually pretty advanced. However, the development of that technology really required a situation where Bronze was no longer a viable solution. In some places, this occurred earlier thanks to them not having access to tin in the first place, and in the Near East this occured likely at least in part thanks to the trade disruption from the Bronze Age Collapse.
Now, I mentioned that the Iron Age really allowed the equipping of larger armies, and this is the real advantage of Iron: it is even more common than copper; once you know how to smelt out the metal from it's ore of course. Once the technology existed to create the temperatures required, suddenly you had an abundant source of material for the crafting of weapons and tools. And ultimately, if one asks what would happen if an army equipped with Iron fought an army equipped with Bronze, it depends: were they similarly equipped? Because functionally, the differences between the metals are not that far off. Certainly not enough to decide a whole battle. On the other hand, an iron equipped army is likely going to have more men who are heavily armed and armoured.