The only way to fix this would be to barrow exactly enough to cover the difference. In the past, Nixon tried to impound money that Congress wanted him to spend, but the court shut it down. The Supreme Court told Clinton that the President only has limited ability to cut spending. Anything Obama or any President does will violate some sort of law.
The idea that one needs to override a debt ceiling is rather unsavory, but without the override, disastrous consequences would take effect. A government shutdown would take place, resulting in hundreds of thousands of layoffs and a drastic decrease in GDP. CBO estimates that if the sequester had occurred on Jan. 2nd, there would be a depression, thus creating even MORE debt due to the lack of taxes in horrid economic times. All of the “no new debt limit” speeches are nothing more than white-knight nonsense, lacking in “Realpolitik” of today’s era.
Should President Obama override the debt ceiling? Absolutely not. It is not for one person to make a decision that will affect every man, woman and child in the country, even if that one person is the president. There have been many proposals and suggestions as to ways to alleviate the weight of our country's debt. Simply raising the ceiling won't do it. If any family in this country accrued debt in this fashion it wouldn't work and they would find themselves in deep trouble. Our country needs to set an example for her people by keeping debt down and balancing the budget, just as each individual is expected to do.
I don't think any one person should be able to make a decision for the entire United States by themselves, even President Obama. Just because the debt ceiling is needing to be dealt with in a very urgent manner that does not mean that other members of our government should not be involved in the decision.