Here's what I would do. But first, let's backtrack just a little. A shut-down government for a few days isn't really going to hurt anything (actually it's a government slowdown, not a shutdown). If some liberal hair catches fire, then there's some entertainment value. So far, the House bill offered a one-year delay of Obamacare (and some other things) and it went nowhere. What would I do next? Play a round of golf, drink a Scotch or two, then propose the removal of taxes on medical devices and "Cadillac" health plans, which were stupid and counterproductive taxes to begin with. Their only reason for being, in my opinion, was for Obama to claim that Obamacare would not result in a net increase to the deficit, and if it takes some dumbass taxes to make it happen, then so be it. When Reid and Obama again refuse to negotiate, I would play another round of golf, quaff a couple of single-malts (look, I'm trying to put myself in the Speaker's shoes here), then fold and put forth a "clean" resolution, which would hopefully occur by Thursday or Friday. I don't want to go longer than that because past elections have consequences, and present actions (or inactions) will have consequences in 2014.
The practical result? The GOP can say that the Democrat heels are so dug in on Obamacare that the law's defenders would rather shut the government down and refuse to negotiate any alterations on any substantive aspects of that legislation. See, it's okay if Obama unilaterally changes Obamacare by delaying its implementation and inserting other niceties (which is constitutionally on thin ice, by the way), but if Republicans try it? Fergetaboutit. Does this mean that I agree with the shutdown? No. The GOP could have easily set a political marker without going this far. Right now, they're pounding on a nail that's already been hammered in. But if this goes beyond the week, then there will be more blowback than benefit.
UPDATE: Derek Thompson has an interesting perspective. Even with Obamacare implemented, the GOP has prevailed on controlling spending. The Tea Party non-compromisers need to look up from their knitting and realize that they've already won on spending.