The Last Option
The validity of the public debt of the United States, authorized by law, including debts incurred for payment of pensions and bounties for services in suppressing insurrection or rebellion, shall not be questioned. But neither the United States nor any state shall assume or pay any debt or obligation incurred in aid of insurrection or rebellion against the United States, or any claim for the loss or emancipation of any slave; but all such debts, obligations and claims shall be held illegal and void.
Nevertheless, the 14th Amendment argument probably is stronger and using it (with a complete explanation to the public) to prevent an outcome desired by the hard right would be quite delicious, given obsessive references to the Constitution by Tea Party acolytes. And the Administration's position likely would prevail, Harvard law professor Lawrence H. Tribe contending "This is not a circumstance in which the courts have any plausible point of entry” and Yale law professor Jack M. Balkin "This is largely a political question. It is unlikely courts would decide these questions.”
A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.
In case Mahmoud Ahmadinejad develops that nuclear weapon he dreams of with the necessary delivery system, and drops it on Chicago, Illinois, be reassured that the Illinois National Guard stands ready to retaliate. Or if a catastrophic terrorist attack were launched on, say, New York City, that well-regulated militia would be the last line of defense for American citizens. And be sent out to hunt down Osama binLaden.
The idea that the Second Amendment applies to modern society is nonsense. But it is commonly held nonsense, in contrast to the strong possibility that Article 4 of the Fourteenth Amendment applies well beyond the Civil War. So, Mr. President, if you can't get an agreement by August 2, follow the advice of Clinton 42 and demonstrate leadership. Challenge the House to impeach you (not likely) and the Senate to remove you from office (a near impossibility). Or dare Congress to file suit and the Supreme Court to stop you.