Can't We Just Go All "House of Cards" on These Guys?

Democrats have their differences, but in the current standoff with Republicans--a standoff that has led to the wholesale shuttering of the federal government--they remain largely united, if not unanimous:  They remain determined to hang together to deny GOP hardliners any semblance of victory.  The Republicans, however, are not so united.  A number of Republican congressmen in both the House and Senate want this shutdown to end.  They know that the longer this goes on, the worse they and their party look: The American people, on the whole, are savvy enough to know that the Republicans forced this crisis, and whatever negative effects are felt, the GOP will shoulder the blame--Tea Party protestations about how President Obama could end the shutdown--if he would just, y'know, cave in to every farcical demand of the lunatic fringe--aside.  And the fact of the matter is that Speaker of the House John Boehner could end this impasse immediately, if he simply called for a spending vote stripped of any extraneous ideological demands--like the repeal of the Affordable Care Act or a recount of the 2012 presidential election.

Boehner, for his part, has put himself in a no-win situation.  He has to know that, in the end, he will have to cave.  He knows that President Obama can not now negotiate with the GOP--even if on some level the President agreed with one or two of the Republican demands--because that would effectively spell the end of the American presidency--not just the Obama presidency, but the presidency itself, as Obama would have established the precedent that Congress can force the repeal of any legislation it disagreed with simply by throwing a temper tantrum and refusing to do its job until baby gets its ba-ba.  John Boehner, in other words, is royally screwed.

So here's my question: Couldn't a group or Republican House members who want this crisis to end--and there are such people--get together and decide on a new Speaker?  Then they go to the Democrats and say, "Look, we know you'd like to have a Democrat in the position.  That's not going to happen, but will you guys all support Rep. X here, so that we can at least get the government up and running again, and then we can thrash out legislative issues in the forum in which such issues should be decided?"

I have no idea if such a gambit could work, but at this point, isn't it at least worth considering?  Maybe the GOP could get Kevin Spacey to make their case.