It is the path to sure defeat. The conservatives already believe the Republican leadership is going to betray them. The house leadership has to force the Dem Senate to vote things down or the president to veto. Otherwise they lose the active support of the Tea Party and won't have a majority in any of the branches next election. Which by the way, also means they will lose the Supreme Court on the next appointment.
Whether the elite rulers in the Republican party want to admit it or not, the Tea Party will turn the next election; either through their support of the Republicans, bloody in-fighting within the Republican Party or as a third party. Only one of those alternatives results in a conservative victory and a conservative Supreme Court.
So you're proposing a shut down. Please outline the steps subsequent to that that will lead to victory on the budget.
Here's what I see: Partial shutdown of gov't services, then three weeks in, public opinion turns sharply against the Republicans in Congress, aided and abetted by the Pravda media. Then , another temporary budget with less cuts in it then they would have gotten otherwise, and continuing hatred towards the Republicans, who begin to buckle individually. Finally, a 5 month budget comes out with no more significant cuts, and some BS talk about a balnced budget. In 2013, a fully Democrat gov't gets voted in..
What fun it is is to stand on principle when you have little leverage!