Here's the text of the Cut, Cap, and Balance legislation Republicans will vote on today. You'll notice that the problem with the Balanced Budget Amendment is dealt within a tidy way.
(a) In General- The Secretary of the Treasury shall not exercise the additional borrowing authority provided under subsection (b) until the Archivist of the United States transmits to the States H.J. Res. 1 in the form reported on June 23, 2011, S.J. Res. 10 in the form introduced on March 31, 2011, or H.J. Res. 56 in the form introduced on April 7, 2011, a balanced budget amendment to the Constitution, or a similar amendment if it requires that total outlays not exceed total receipts, that contains a spending limitation as a percentage of GDP, and requires that tax increases be approved by a two-thirds vote in both Houses of Congress for their ratification.
So there are options. The ultra-specific versions of the amendment are actually cited, but in the world of theory, Democrats could agree to another text and satisfy the GOP. Well done. No one's done the legwork for a new BBA, but this makes it tougher for the White House to dismiss it.