Yes. That's what I thought you wanted to do, when you wrote:
The word 'alternate' and the line about ancient Egypt ruling 'a pretty large chunk of the world' made me think that you wanted to change history. You asked where to start, so I suggested considering when specifically your time-line could be different from actual history, and why and how it would be different, and whether that difference would produce the outcome (a more powerful Egypt) which you wanted to see.
You're considering writing about when ancient Egypt was at the height of its power. That sounds like around the 15th century BC, in the New Kingdom period.
You wrote the 'Hittites are already conquered'. I'm nore sure if you're indicating the time when you'd like to write your story (if this refers to the story being set after the war of Ramasses II against the Hittites, that would suggest a date after 1258 BC) or if you are saying that this could be how your history differs from actual history. Are you saying that, in your ancient Egypt, another pharaoah fought the Hittites at an earlier date and more successfully than Ramasses II? If so, you could think about how that pharoah would have succeeded where Ramasses II failed. For example, historically Ramasses II failed to exploit his victory in the
Battle of Kadesh. You could work out how a pharoah could have succeeded where he failed.
Obviously, in the history of our world, ancient Egypt declined after reaching the height of its power. I wonder what you see as the causes of that decline and how your ancient Egypt will avoid that (if it will)?