On April 15th, 2011, The President signed a new law called the 2011 budget. That law implicitly repealed the "debt limit" law passed by the PREVIOUS LEGISLATURE. The previous legislature cannot restrict the law making of the current legislature. Such a constraint is know as "legislative entrenchment". The "debt limit" law is unconstitutional for that reason. You can see that the congress passed a budget that was in violation of the "debt limit". That simply means that the "debt limit" has been superseded by the new legislation.
This SHOULD BE the position that the president takes. As it is not possible to "faithfully execute" the two laws then the president must choose the latter law as opposed to the former. As such, the president is OBLIGATED to issue an executive order to ignore the superseded "debt limit" law. If congress wants to pass a NEW "debt limit law" then congress is at liberty to do so, and Barack Obama MAY BE at liberty to veto it. Last year's "debt limit" law has been implicitly repealed by the 2011 budget law.
What actually matters here is the prerogatives of the president. If the House does not like the president's executive orders then the House can impeach the president or file suit in the courts or create and pass a new debt limit law that passes both houses of the congress. It seems to me that the president has no real choice in this matter. The Congress has failed to create non-conflicting laws and the president is forced to "make law" because of that fact. According to the War Powers Act, a joint resolution of the congress is all that is necessary to override such an executive order. If the War Powers Act is correct then the president cannot use the veto power to prevent a congressional override of his "executive orders", military or not. Let us be clear on this: A national emergency that will harm the union is to be addressed by the unitary executive and that is why we have a unitary executive and NOT a council of lords. The congress can override an "executive" order by joint resolution BECAUSE the congress and NOT the president has the law making power and the war declaration power. The veto (according to Jefferson and others in the long line of presidents) should only be used to strike laws that would weaken or violate the constitution. The president's executive orders are only issued in cases where there is no clear path to the preservation of the union other than by executive order. Congress can easily correct this lack of clarity and override the executive order if congress so choses.