Despite the deep divisions in Washington and the looming election, Congress still has at least one job to do before November 6. They have to pass a budget or at least fund government operations when the next fiscal year begins on October 1. House Republicans have not shifted their view, regardless of the popularity, that they must destroy the Medicare system in order to save it, through a privatization scheme that would end the guarantee for single-payer Medicare. Democratic Senator Ron Wyden handed Paul Ryan a lifeline by signing up for his newer premium support plan, which wouldn’t eliminate traditional Medicare, but which would put it in competition with private options, and provide a voucher to seniors to make the choice themselves for their plan. Nancy Pelosi called that solution “lipstick on a pig” late last week, arguing that the plan would “break the Medicare guarantee” and that Wyden’s presence on the bill would not deter House Democrats from criticizing it.

Despite this, House Republicans are likely to include that Medicare premium support plan in their budget resolution. But the larger question, Roll Call reports, is whether any budget resolution can pass at all.

House Republican leaders have been holding listening sessions with their Members on the topic, but they are in a bind. It is unclear that they have the votes to pass a budget resolution with the numbers from the Budget Control Act, as many conservatives believe it would spend too much.

Lowering the spending cap, on the other hand, would leave appropriators with unrealistic numbers and could doom any spending bills to failure on the House floor.

Not passing a budget is also undesirable for the House GOP, as it has pilloried Senate Democrats for deciding not to pass a budget resolution this year and for relying on the Budget Control Act.

The right flank of the caucus wants to go below the discretionary spending levels set by the spending cap in the debt limit deal, about $1.047 trillion. But that doesn’t even take into account the trigger cuts to discretionary spending, which will chop close to $100 billion more. Many Republicans want to restore the defense side of that trigger, but not the discretionary side.

But this is where the Medicare plan comes in. If Ryan adds that piece into the budget, he’ll get almost no Democratic support. But if he keeps the baseline spending at debt limit deal numbers, he’ll lose Republican support on the right. And if he reduces spending below the cap, the House might pass it but the Senate will reject it. It’s hard to see any agreement on any side for the budget.

Which means we’re probably looking at a short-term resolution, perhaps into the lame duck session. And we know that there are already a host of issues looming in that lame duck, including the Bush tax cuts, the payroll tax/UI deal, and the trigger. Throw the FY 2013 budget into that mix.

The one caveat to this is if Republicans feel pressured to pass an on-time budget, after criticizing Democrats for years for failing to do so. If that’s the case, we could see a dynamic similar to the payroll tax deal, where Republicans caved to get the issue off the table. But it was easier to do that on tax cuts than it will be on these momentous budget issues, where Republicans are in ideological lockstep.