I really don’t think we’re stupid enough to try that.
Logistically we’re already strained from Iraq.
Politically it’s a bad move to invade one of the region’s only democracies. Obviously it’s a weak democracy which is why the elected reformers haven’t made much headway, but an invasion would undercut them and turn a mildly pro-American population against us for years to come.
Finally, Iran could make things very difficult for us in Iraq if it wanted. So far they haven’t interfered much, either confident that a pro-Shii government will be elected, or divided within their own government on the issue. This reticence would evaporate upon US aggression.
Having said that, should the US choose to engage in such adventurism, I expect that it will be quite limited - going in, occupying a few towns for a week or two, with various special forces missions throughout the country, and then disengaging. If we can disengage at that point.