No way! Bush would never go for it.
What I have heard from behind the scenes is that another several months of no government in Iraq and the US would have been willing to sit back and let the pro-Iranian Shia be kicked out. The US no longer considers Sadr and a number of others like Jaafari in the UIA to be democratic or willing to allow for future free and fair elections.
There isn't democracy as you or I know it in Iraq, the elections in the south of Iraq were as rigged as the Iranian election. The US knows it and has used the evidence they collected of the militias and Iranians rigging the vote to force the UIA to promise to make a government of national unity.
Right now the political process finally has a chance to move forward if the UIA makes a decient choice of PM. If the UIA choses someone as bad or worse then Jaafari the political process as we know it will be stuck in limbo and every second the political process is stuck in limbo with no government is a second Iraq moves closer to the edge of the cliff.
A coup would be far preferable to Iraq decending into endless civil war and chaos. Allawi actually does believe in democracy, but at the same time believes the Iraqi state is on the edge of the cliff. It is now up to the UIA to choose what future Iraq will have.