Usually I don't have a clue on what should be done on iraq, here are some ideas though.

Divide the country into 3 separate ethic areas, like states in a federation as close as possible according to ethnic lines.

Have local elections for each town and 3 congressional bodies with either a president or leader of each part like a Prime Minister. Each town elects someone to the state parliament. Have a constitution that applies to all 3 states.

In any decision that affects all 3 states it must be decided unanimously. For example with 3 presidents all 3 have to agree to stop 2 states ganging up on the other. This works in the European Union where in many issues all states have to agree.

The solves the problem of three different people in one country, which was created by Britain when they drew the borders of Iraq. Technically creating these states violates the Geneva Convention but they could be ratified by elections or a referendum. When each group saw the benefits of self rule as opposed to rule by others then they would likely ratify it.

There would be a common army directed only be a unanimous vote of all 3. Police would be a separate state matter. The federation would handle monetary policy with a single currency with all 3 states sitting on the reserve board and again needing a unanimous vote. Oil would be a national resource and proceeds divided equally according to the population from each state. All oil policy must be unanimous for all three states. All would have a right of passage and to settle in another state if they desire.

There are plenty of exampes on how this has worked well in the past. Yugoslavia was stable for a long time as was Czechoslovakia. What will probably happen is one or more states will eventually become taken over by a religious party or secular Baathists but the need for unanimous voting will prevent the whole country going this way.

The US can court relations with individual states playing them off against each other in offering aid. The other states as they miss out will tend to moderate.