Yeah I also think the debt ceiling may not be the place to pick a direct fight. We are talking about cutting off spending that Congress had already approved, which is too easily demagogued as "refusing to pay our bills." The point is to limit the bills Congress racks up in the future.
My suggestion is that the responsible members of the House declare that they will not vote for any spending beyond the previous year (or some previous year), adjusted for inflation and population growth. This would still be a substantial "cut" according to the sophistry by which increases in spending are projected and anything less that the projection is a "cut." It would be a defensible line in the sand. Then hold firm.