Obama should go all in and leave the Republicans to agree or have the cuts. He has nothing to lose, he can't be re-elected and almost everything he would want to do with this term will be shot down by the House anyway.
America responds well to a strong President even if he's wrong. By compromising, this term and last, he's looked weak and lost all his political capital.
That said even Obama's best hope for a bill that will pass is still 0% of the way towards actually reducing the deficit on a reasonable timescale.