Obama Should Throw the Republicans a Bone, then Declare

NYT:

The nature of Speaker John Boehner’s final battle with the White House on the budget crisis is now clear: It doesn’t matter what House Republicans win in exchange for raising the debt ceiling and re-opening government, as long as they win something...

...And that’s precisely why the president can’t agree to it, even though the impact would be minimal.

I'm extremely sympathetic to this argument, but I have an idea for an alternative. Obama gives the Republicans the smallest, least harmful thing that will provide the fig leaf they seek. Then, as soon a the dust settles, he explains what he has done. In plain language, explains how the Republicans sausage was made.