I still don't know if we should have gone into Iraq, at least not until after all the bad guys in Afghanistan were dead. But if we had, then what we should have done after Saddam fell is to correct the errors made at San Remo and partition Iraq into three countries--Kurdistan in the north, Shia Mesopotamia in the east, and Sunni Iraq in the west. This was actually Joe Biden's proposal (like a stopped watch, Crazy Joe may be right from time to time), which may explain why the neocons in the Bush administration were opposed. The dividing lines are pretty straightforward:
The stumbling block to Kurdistan was Turkey. But there was a solution. Back then, the EU was worth joining, and Turkey really wanted in. So give Schlumberger the Halliburton contract for Kurdistan, tell the French that this is the makeup call for what was done to them at San Remo, and it's their job to convince Turkey that Kurdistan is the price of EU membership. And besides, Turkey, to the extent your Kurds decide to be patriotic and relocate to "their" country, that solves a major problem for you.
I would have left at least some of the old Ba'athist power structure in place in Sunni Iraq. The elimination of that power structure was one thing that led to a lot of instability in postwar Iraq, and IMO ultimately to the rise of Daesh/ISIS. The Ba'athists were mean enough to keep things in check. My guess is that they would have allied with the Sunnis in eastern Syria, probably with a lot of Saudi support, and eventually would have taken a lot of eastern Syria away from Shia (Alawite) control.
I don't think there is a way to get there today. We have pretty much totally screwed the pooch in our handling of the region. Maybe it's time for us to quit screwing it up. If we pull out, things will deteriorate, I'm just not sure how far. As long as, "My brother and I will fight my cousin, my cousin and I will fight the world," remains the mantra, there is going to be instability. But maybe it's time to let China handle it--if they can. They're the ones who can't survive without the oil. If it distracts them as much as it has distracted us, that would be a good thing.