This is a complex and in some ways contradictory moment for congressional Republicans. They just won a significant victory by gaining control of the Senate and holding on to the House. The president they’ve opposed with such vigor is in the autumn of his term and has weak approval ratings. And yet much as this should be a moment of triumph for them, it must feel like they face nothing but frustration everywhere they turn.

For starters, because President Obama still wields a veto pen, they can’t actually pass the kind of legislation they’d like to. Tax cuts? Rollback of environmental regulations? Dismantling the Affordable Care Act? It’ll all have to wait — and if they can’t manage to win the White House in 2016, it’ll be waiting for a long time.

And of course, victory hasn’t tamped down the fundamental division in the party they’ve been wrestling with for the past few years, between tea partyers who want nothing but maximal confrontation and the more practical-minded members who want the GOP to be known for something other than creating havoc.

And while Obama may be weakened, he still has the ability to back them into a political corner, as he’s done with immigration. All the options Republicans face on the issue are bad ones. Fight him too hard and they reinforce their problems with Latino voters; shut down the government to stop him and they risk political catastrophe (even if it’s only temporary). But if they don’t fight him hard enough, there will be an internal rebellion.

Here’s their latest attempt to find a way out of that conundrum, via Politico:

House Republican leaders are beginning to coalesce around a strategy to avoid a government shutdown in less than a month.
The likely proposal would fund nearly the entire government through September 2015, but immigration enforcement related funding would be renewed on a short-term basis, according to several high-ranking GOP lawmakers and aides who described the plan as it stands now.
The strategy is designed to keep the government open, while satisfying the base, which is livid with President Barack Obama for issuing an executive order that ends deportations for millions of undocumented immigrants.
Republican leaders have struggled to come up with a plan that would satisfy both goals — of keeping the government open, while allowing members to express their anger at Obama. GOP sources believe that keeping immigration funding on a short leash could be the answer.

The only trouble is that the agency that will actually be carrying out Obama’s executive actions, U.S. Citizenship and Immigration Services, is funded by user fees, which means it doesn’t need money from Congress, which means Republicans can’t shut it down by refusing to provide funding. So it’s unclear exactly what sort of immigration enforcement funds Republicans are contemplating withholding.

This may not be the last proposal congressional leaders come up with — in fact, I wouldn’t be surprised if they have to go through a few more before they find one that the tea partyers can live with. What that suggests is that the bulk of Republicans’ time and attention in the coming months will be spent on managing this internal conflict — satisfying their base, keeping intra-Republican arguments in check, and trying to do it while finding effective ways to stymie the administration.

But it’s unlikely that they’ll be particularly effective at that last goal if they’re constantly looking within. Today we also learned that Jonathan Gruber will be testifying before the House, which I guarantee is something that will have lots of members excited. In some ways it’s the perfect microcosm of the GOP’s current status. Republicans will go in sure that they’re really going to stick it to Obama this time, do a lot of gavel-pounding, and find that they accomplished little but giving conservative media something to talk about for another day.

Meanwhile, Obama will still be president and will be spending his time doing all the things that keep making his opponents mad. While they’d certainly prefer having majorities in Congress to being in the minority, it’s going to be a frustrating time for Republicans.