I think c. 1200-1100 bc is the earliest Edom could have really had a serious kingdom established. By 1200, Egypt and the Hittites had largely diminished in influence over Palestine and transjordan. But the area of Edom was indeed a cultural center for nomads & Shasu long before the Egyptians controlled palestine.
The people of Israel established their kingdom around 1020 bc, and it seems very plausable that Edom and Moab established their kingdoms 100-200 years earlier, amidst the shrinkage of Egyptian power.
The biblical tradition of wandering Israelites passing through the kingdom of Edom c. 1200's bc isn't really that problematic if Edom was just beginning to form it's monarchy at this point, but would have probably been a coalition of powerful tribes.
Around 1200-1100 bc, all sorts of foreign peoples and nations began to stake claims for their share of greater Palestine. Philistines,Amorites, Canaanites, Moabites, and Edomites would have all carved out their portions of the land. And on top of this we have the Israelites and Hebrews who are another addition to this new era.
It's probable that there would have been more than one exodus of Hebrews and slaves from egypt during this time period of Egypt's decline.
Palestine was constantly under rural warfare even from the beginning of Egypt's 400 year dominance. In the 1400-1300's there were raiding parties of Shasu and Hapiru who attacked the cities of Canaan. many of these Hapiru and Shasu were captured and sent back to Egypt as slaves
(literally thousands at a time). By 1250 under Rameses rulership, Egypt was already beginning to Weaken, and Merenptah's campaign into Palestine had little affect, though it may have offset Israel's development for a generation or two.