"How Are You Going to Pay for This?"

Once the current "debt ceiling" pseudo-crisis has passed, assuming he is not able to achieve a resolution that takes the debt ceiling out of the picture indefinitely, why wouldn't the President be able to ask, in response to any new spending or budget, "How are you going to pay for this?" If the answer is "spending cuts", the President can ask that they be included in the bill. If there is no answer, the President can reply, "Then as part of this bill you need to either find a way to pay for your spending, or you need to increase the debt ceiling to cover the debt that the nation will incur as a result of this spending."

The Republicans could attempt some blackmail or brinksmanship, "We absolutely refuse to fund this bill, that is essential to our national defense", but I'm not seeing how they would be helped by taking such a position. If the President agrees that the matter is so essential that the spending must be approved despite its creating a debt ceiling issue, the President can make a clear statement about what's happening (call it "public shaming") and the Republicans will should pretty foolish if they later refuse to increase the debt ceiling to pay for the spending they deemed essential.