There's a whole other option I haven't seen discussed that can provide a way out of this mess. One that involves both the 3rd branch of government, and We the People. And that is:
The Fourteenth Amendment says:
"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."
If, indeed, the debt ceiling is possibly unconstitutional under the Fourteenth Amendment, as many have recently supposed, then we do NOT need to wait for some government official to say so. Any federal court in the United States has the power to enjoin the enforcement of an unconstitutional law. And any citizen who will be harmed by the failure of Congress to raise the debt ceiling (and this is everyone in the country, maybe in the world, but especially recipients of Social Security payments, veteran's benfeits, Medicare payments, government employees to whom pay is due, and holders of U.S. Savings Bonds) can go out tomorrow and FILE A LAWSUIT in their nearest United States Federal Court seeking an EMERGENCY PRELIMINARY INJUNCTION against the enforcement of this unconstitutional law. If a hundred lawsuits are filed in a hundred courts, then it only takes ONE judge of all of those lawsuits to decide that the case has sufficient merit to issue a preliminary injunction against the operation of the law.
A few notes (based on convo with a far more knowledgeable friend):
Standing: It may not be clear who has legal standing to bring the suit, but if many people with a wide variety of claims file lawsuits, it is much more likely that some courts will find some of them to have standing.
Defendant to be sued: It also may not be clear who the suit should be filed against, so the best bet is to sue everybody, including the United States as a named party, but also name the President, the United States Department of the Treasury, the Federal Reserve, and Congress. If there are more potential defendants that I'm not thinking if, sue them too. The worst that can happen is those defendants will be thrown out of the lawsuit.
Court to sue in: Lawsuits should be filed in as many courts as possible, but at least several should also be filed in the U.S. Court of Federal Claims, which deals with "takings" (where the government takes property, and the injured parties are entitled to due compensation under the Fifteenth Amendment), as that may ultimately be decided to be the "proper" court. There may be less chance to find a state court that will consider the case, but then again, it couldn't hurt to file a few suits in state courts.
Also important: When filing a lawsuit on this ground it is important to tell the Court in the complaint that you have standing because you will be injured by the statute, to state specifically what loss you personally will be facing (social security, veterans benefits, your government paycheck, the ability to redeem your bonds) and to specify the statute (it's been suggested to me that it is likely the Second Liberty Bond Act of 1917, P.L. 65-43, 40 Stat. 288, enacted September 24, 1917, Currently codified as amended as 31 U.S.C. § 3101; or the Public Debt Acts of 1941 (P.L. 77-7), 1942 (P.L. 77-510), 1943 (78-34), 1944 (P.L. 78-333), and 1945 (P.L. 79-48). To be sure, just list them all.
No need to take this lying down. The citizens can get involved immediately if they choose, and ANY injunction by the court would instantly lift the burden from the President and Democrats to agree to any draconian, nation-damaging, de-stimulating, based-on-fantasy spending cuts package of the vital entitlement programs many American citizens rely on to stay alive. And to the world, it would appear that our 3 branch system is being employed, and thus, even if the House is clinically insane, at least America still has courts to act as a check and safeguard the nation.
Additionally, a huge public display by hundreds, even thousands of citizens filing suit could bring attention to the people who are being most directly held hostage by the Republicans terrorist tactics. No doubt there are decorated veterans and very empathetic people who would be affected and would make a great public face for the harm being done - and some may have stories the media would be eager to cover.
And if ultimately the Supreme Court decides in a 5/4 vote that the debt ceiling is in fact constitutional, President Obama could still then order the minting of a $3 trillion coin and be done with it. Even then, he still has the 14th Amendment.
Of course best of all would be if someone (Mr. President) would actually debunk the supply-side/Milton Friedman/Ayn Rand drivel in the first place, then we wouldn't have to keep suffering through the carnage of horrible economic policy, and be constant hostages on account of the authoritarian followers of Hoover/Reagan/(Obama?)nomics!