A better strategy for House Republicans to force some spending cuts on a President who doesn’t want them looks like this.

Rely first on the sequester, second on the CR, and last on the debt limit as leverage.
On the debt limit, state loudly and repeatedly a simple principle: We will pay our bills on time and we will cut future spending. Rather than being against a debt limit increase unless it also cuts spending, say that you’re for a debt limit increase that also cuts spending. You’re the legislative branch, you control what’s in the law. Act as leaders doing the right thing, rather than as rebels trying to block President Obama and Congressional Democratic spenders from doing the wrong thing. Agree with the President that we must pay our bills, and politely smile and say “And we’re going to cut spending, too.” Make him argue against cutting spending, rather than giving him the opportunity to attack you for risking financial disaster.
Pass a bill out of the House that raises the debt limit and cuts spending. Make it a short-term extension, maybe 3-6 months, and cut spending by a similarly modest amount.
Take whatever big spending cuts you want and make them conditions of extending the Continuing Resolution. Threaten to shut down the government rather than to make the government risk not paying its bills on time. It’s a less damaging and therefore more credible threat.