Call Boehner's bluff
For once I'm with Jolly Roger on this one. It's time to call their bluff. Demand a budget based on economic reality. It's what the people want; poll after poll shows that Americans overwhelmingly want the deficit reduced by restoring normal tax levels on the wealthy and corporations, not by spending cuts. There are signs that House Democrats are ready to take a tough stand. If Boehner really tries to vote down the debt-ceiling increase (which is far from certain, no matter what he's saying now), at least some House Republicans would probably defy him and defect, especially with Wall Street warning them that even talking about voting it down is too dangerous to risk. If it gets close, specific concessions could be offered to individual waverers.
And if Boehner really does manage to stop the debt ceiling from being increased? That would likely cause a planet-wide economic crash, but the concessions he's demanding now would do terrible harm to our economy, driving unemployment and inequality to even more ruinous levels. And giving in to blackmail in this case would just encourage more blackmail down the road. The focus should be on taking back the House next year, so the government can govern without economically-illiterate bigots holding a gun to its head.