Only Revenue Bills are required by the Constitution to originate in the House.
What the Appropriations Committee needs to do is refrain from passing any omnibus spending bill. They need individual spending bills for each department or agency. They should be frugal bills and the should speed them over to the Senate. Let the Harry Reid be the one stopping the bills in the Senate or make Obama veto them. The house should pass only the required spending.
That is exactly how it must be done! An omnibus spending bill allows the president to "shift" the uses of that whole pot of funds. But if Congress passes individual bills allocating limited funds for specific departments and agencies, the president can only spend those amounts toward the uses for which they were appropriated.
The House of Representatives holds all the cards, since it alone may originate spending bills. What remains to be seen is whether or not the House has the guts to play its hand. Whig status awaits if the GOP folds...