The party leadership, including House Speaker John Boehner and incoming Senate Majority Leader Mitch McConnell, desperately want to avoid a shutdown either this month or in the next Congress to show that they can govern effectively and to move on such issues as tax reform, trade deals and bills to make changes to Obamacare.
But GOP firebrands like Rep. Steve King of Iowa and Sen. Ted Cruz of Texas are demanding immediate and tough action against the president over immigration and say a shutdown should not be taken off the table.
And they can use the results of the midterms to say going hard after the president—including by shuttering the government—is not a losing strategy, as the GOP leadership believes.
Read MoreThe big risk of GOP vs. Obama immigration battle: A shutdown
"At the time you and a lot of folks in the press said what a disaster it was to stand up to Obamacare," Cruz said on "Fox News Sunday" about the last government shutdown. "Let me point out, we just had a historic election where we won nine seats in the Senate, we retired Harry Reid, we just got the biggest majority in the House since the 1920s."
Cruz and some other Republicans want to pass a continuing resolution funding the government that cuts all money for agencies that would enforce Obama's immigration orders, including the Department of Homeland Security.
Obama would surely veto any such measure, setting up a possible shutdown confrontation, which could once again put a dent in consumer confidence and the economy.
Even if the Republican leadership manages to corral members into voting for a "clean" continuing resolution after Thanksgiving that punts the issue to the next Congress it will just ensure that the GOP will not be able to move on from the issue and will have to revisit it early next year.
Read MoreHouse Republicans Sue Obama Administration Over Health Law
In fact, a CR that only lasts a couple of months could push the issue up against the spring deadline for raising the nation's debt limit, another lever conservative Republicans are intent on using to wring concessions out of Obama and Democrats. A spring fight that wraps in immigration and overall government spending and ties it to raising the debt limit and keeping the government open could once again damage a fragile economy and scare markets that are already susceptible to any moves by the Federal Reserve to begin its eventual campaign of rate increases.
At the heart of all this is a deep struggle within the Republican Party over the issue of immigration reform and whether the GOP should be presenting a forward-looking governing agenda or exist mainly to oppose Obama.
Many House Republicans feel they owe their majority to fierce opposition to Obama and do not want to give him any victories on immigration or even keeping the government open and functioning. Senate Republicans also feel the pull of fighting Obama but see their 2014 wins in large part as the result of voters being deeply frustrated with a dysfunctional government and a desire to see the parties come to agreements that can help the boost economic growth and lift wages.
These Republicans point to surveys like that latest Wall Street Journal poll showing 63 percent of Americans want the parties to work together to broker deals rather than stick to rigid ideological lines.
The differing views are already apparent in the field of potential 2016 GOP nominees as well. Sen. Rand Paul, R-Ky., for instance, has joined Cruz in demanding a swift and firm response to Obama on immigration, though he has stopped short of saying what exactly that response should look like.
Read MoreObama sabotaged immigration reform efforts: GOP
Former Florida Governor Jeb Bush, by contrast, also slammed the president's unilateral move but said the answer is for the new Republican Congress to pass its own big immigration reform bill and get it to the president's desk.
Whether Republicans tend toward the Jeb Bush model—oppose Obama by governing and sending legislation to his desk—or toward the Rand Paul/Ted Cruz model—oppose Obama by whatever means available—will determine how scary a year 2015 will be for markets and the economy.
—Ben White is Politico's chief economic correspondent and a CNBC contributor. He also authors the daily tip sheet Politico Morning Money [politico.com/morningmoney]. Follow him on Twitter @morningmoneyben.