The President has the power to sign or not sign legislation which is sent to him. If the legislation is Constitutional as the President sees it, then he may choose to sign, creating a law, or he may choose not to sign, and no law is created.
If the legislation is unConstitutional, then the President is obligated by his oath of office to not sign it. Whether he is a lawyer or not is irrelevant. His oath obligates him to preserve the Constitution. The intention of the oath is to make clear that he must judge the legislation. He has resources to hire many lawyers to make his decision.
When an Attorney General, who is picked by the President, is faced with prosecuting an existing law, the President's oath and the Attorney General's oath require them to exercise judgement because the duty includes preserving the Constitution. Because we have a system of appeals which can take any case to the Supreme Court, the President and the Attorney General are free to prosecute with the expectation that the Supreme Court will throw out the law.
Courts routinely allow "amicus" briefs from "friends of the court" so that opinions which do not align with the prosecution or the defense may be heard. The President and Attorney General have this or a similar mechanism to make their own opinions heard. This mechanism should be sufficient to address any doubts which the President and Attorney General have. When the Supreme Court has decided, then the President risks impeachment if he does not agree.
The only unworkable part of this system is that the unConstitutional aspects of our government have grown so large that it takes a number of years for a case to reach the top court. During this time, a defendant can rot in prison or some other injustice can go unaddressed. This is because so many Presidents before have signed unConstitutional bills.
The "death spiral" created by unConstitutional laws too numerous to address by the Supreme Court is where we stand today. That is why Bush has an obligation not to sign additional unConstitutional measures.