If the Republicans want to pivot from the Bush tax cuts to a fight over raising the debt ceiling, I think that the Obama Administration should be more than willing to let them (although perhaps it should not signal in advance just how willing).
Once a debt ceiling fight starts, I think the response is pretty simple. Scorched-earth rhetoric (phrases such as "blackmail," "extortion," "disloyal," "sick," "perverted," and "economic treason") would not be out of place. You don't really need to be civil with blackmailers who are threatening to set a torch to the U.S. and global economies, and to tarnish for generations the U.S. government's credit rating on world capital markets, if they don't get their way.
Then, if the Republicans pull the trigger, either the 14th amendment option and/or the platinum coin option, both of which have been discussed elsewhere, could bring the whole thing to a satisfying close. Indeed, the main question might end up being to what extent the Republican leadership should be humiliated, rather than being offered a moderately face-saving climbdown. It's probably best to be nice, even though they wouldn't deserve it, and that is likely to be Obama's instinct as well (perhaps even to excess).
Presidents who "stand tall" to save the U.S. from disaster generally benefit politically from doing so. If the Republicans want to respond by impeaching Obama in the House of Representatives, that's their privilege, and they've already shown that they like doing this to second-term Democratic presidents. As I recall, that worked out great for them in the 1998 midterm elections. If they are hoping for low turnout in 2014, along the pattern of 2010 rather than 2008 or 2012, impeachment by the House is an excellent way for them to ensure that they will not get it.
Legally, the issues have been much discussed elsewhere, and insofar as there is any answer (although the normative criterion for devising answers to open constitutional questions is unclear), Obama's position would be highly plausible at a minimum (as I recall, especially on the "platinum coin" option, but really on both). If the five Republicans on the Supreme Court want to unleash a global economic calamity by reaching to strike it down, I suppose they can, but that again would fall on them.