Assuming Pelosi releases her caucus to vote as they wish, then the Bill clearly doesn't need the liberal Dems to put it over the top. It just needs a handful, which Obama/The White House can surely whip up.
Why would Obama and the Democrats want to save Bhoener's skin, again?
They'll get blamed if they're the ones who vote down the bill. I'd vote against it myself but from the perspective of success in the court of public opinion, it isn't in their interest to oppose this.
Because the last shutdown worked out so badly for the Republican Party in this year's elections.