I'm still not sure what Krugman, PM Carpenter and others want Obama to do in the face an opposition that doesn't want to compromise on this issue, but it seems to me that among the reasonable interpretations to be made from GOP rhetoric is that they don't much care if Obama exercises the so-called Constitutional Option--essentially ignoring the debt ceiling restriction and authorizing Treasury to pay the country's bills anyway.
Certainly their drastic-spending-cuts-only demand is ludicrous and not made in good faith. In addition, their offstated claims that the debt ceiling clock isn't a serious matter and the "Treasury has lots of means at its disposal to avoid default" bare the sound of a party that expects its demands to not be taken seriously and an invitation for the President to act regardless of any authorization legislation.
Obviously, at some point, the GOP would rail against the President one way or the other. But their language so far seems, weirdly, to be anti-crisis. And given the tragedy that would result from implementing the GOP's Dream Spending Cuts as an alternative, it's hard to imagine a worse outcome than the Constitutional Option.