There are a few holes in your idea:
1) Obama has shown the willingness to veto partial budget bills, and the Ministry of Propaganda (NBC/CBS/ABC/CNN/WP/LAT/NYT) cheerfully blames “Republicans who are shutting the budget down”.
2) If Obama gets any money at all, he will spend it on things that the money is not earmarked for. Clinton did it, you KNOW this Muslim gay cracksmoker will.
That is why the Republicans must parse the heck out of the budget bills. Imagine how Obama would panic if hundreds of thousands of bureaucratic voices “cried out for money and were suddenly silenced.” Bureaucrats are more than happy to save their own skins by sacrificing their peers.
And “shutting down the government” doesn’t work if it is just a single agency at a time.