So this is not an issue.
Or were you talking about the current government shutdown because of no funding appropriations? The two are completely different issues that just happen to occur 17 days apart.
If Boehner never backs down on the current shutdown, then Obama only has two options:
1. get rid of Obamacare, authorize the Keystone Pipeline, increase offshore drilling, and a few dozen other demands by Boehner, or
2. let the government remain shutdown until 1 January 2015.
But there is a third party to all of this: State Legislatures. Two-thirds (34) of the State Legislatures can demand an Article V Convention for proposing Amendments to the Constitution of the United States. Such a Convention could propose an Amendment along the lines of:
"If no Appropriation has been made current by Law, per Article 1, Section 9 of this Constitution, for the Executive to perform an Action required by Law, the most recent Appropriation for that Action will continue until a new Appropriation is made, or the Law requiring that Action is repealed. If no prior Appropriation exists, the Action will not commence.
"When the Treasury does not have Money to satisfy Appropriations made by Law, the Treasury is implicitly authorized to raise that money through the issuance of Treasury Bills and Bonds."
If passed then ratified by three-fourths (38) of the States, last year's funding levels would continue unabated until funding is changed. The first paragraph would end this and all future shutdowns. The second paragraph would eliminate the debt ceiling for all time.
I wouldn't be surprised if that would alleviate the Congress of 99% of its workload freeing them to get down to the real business of Congress: fund-raising for their next election.