Obama rejected the Fourteenth Amendment option much too quickly. But the path is still open if he can muster the gumption to use it.
If Republicans raised a Constitutional question, the case would have to go before the Supreme Court. In that eventuality a defendant would have to show personal harm – a difficult proposition.
A shrewd lawyer could probably conjure up a ridiculous harm a la Bush v Gore and present the case. However,it would likely be recognized the world over as a purely political maneuver and be rejected by public opinion as such. Obama would have the people on his side – a powerful force if he chose to hammer the opposition with it.
Also, something to keep in mind is that the Court has no Constitutional authority to declare any law or action unconstitutional. That authority derives solely from precedent and an action of this type by the Court would raise questions of separation of powers, particularly in the event the president were to declare a national emergency and claim that he is invoking the powers granted to him by the Constitution as commander in chief.
Moreover, some scholars have pointed to the oath of office of the president and claimed that the oath itself grants the president the power to refuse to obey a law (or a decision of the Supreme Court) if he construes the law as unconstitutional.
The case of Andrew Jackson against the Second Bank of the United States offers some insight into the powers of the Executive versus the Judiciary. Jackson vetoed the law establishing the bank after the Court declared the law constitutional.
And in Worcester v Georgia, a pro-Indian decision of the Court with which Jackson disagreed, the feisty president is said to have remarked that “John Marshall has made his decision; now let him enforce it”. Marshall did not enforce it and avoided a confrontation with the Executive but the affair sheds light on the powers of the two branches vis-a-vis each other.
So Obama has options. What he lacks is the courage to deploy his powers.