Great question. Unfortunately, I had to leave out much that could have been said in this brief article. The first part of the answer has to do with the conclusions one reaches about the likelihood of the Exodus having taken place before the well-documented timing of the battle of Kadesh. Article JBOTL13A summarized the arguments of Rendsburg of the likelihood that the Exodus itself did not take place till the time of Ramesses III (although the Israelite oppression and the building of the pharaoh's cities almost certainly took place under Ramesses II). The second part of the answer has to do with detailed analyses of the two texts and their provenance--if you are really interested in this question, I would heartily recommend going directly to Berman's 2017 book.
I didn't see any reference indicating why the Exodus account followed the Battle of Kadesh account. It seems to me that it could have been the other way around with the Battle of Kadesh account echoing an earlier, embarrassing defeat of the Egyptian army by the Israelites. Maybe I missed something in this article but it seems very likely to me that the Egyptians were the ones who copied an earlier event.
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