Just when it seemed that the Democrats running Congress had given up their losing battle with President Bush over the war, along came his veto last week of an expansion of the federal children’s medical insurance program. While the two issues have nothing substantively in common, they are absolutely connected politically.

The president’s war critics on Capitol Hill are searching for any area of disagreement with Bush that might generate enough Republican votes to override one of his many threatened vetoes — and thereby, perhaps, undermine his clout going into the next Iraq showdown.

To that end, Democrats are launching a two-week public relations campaign to pressure more Republicans to vote against Bush’s veto of the bill expanding the State Children’s Health Insurance Program. Sixty-seven senators voted to clear the bill, precisely the two-thirds majority required to guarantee an override. But only 265 House members did likewise; to assure victory, the legislation’s advocates need to find 25 more lawmakers to join their side.

While Democrats care deeply about providing health care to more poor kids, the SCHIP fight also represents an important moment in their ongoing effort to peel Republicans away from their White House loyalties — in the hope that such a mutiny will encourage more GOP defections in the Iraq debate as the 2008 congressional election draws nearer. And so Rahm Emanuel of Chicago, the chairman of the House Democratic Caucus, wasted no time connecting the two issues when talking to reporters right after Bush issued his veto, accusing the president of seeking an “open wallet” for Iraq but an “empty stocking” for children without medical insurance.

Even if Democrats can’t find the votes in the House to enact the health bill over Bush’s veto, they’re calculating that a victory in at least the Senate might chip away at the president’s overall authority. And portraying Republicans who vote with Bush on SCHIP as “hurting children,” as Senate Majority Leader Harry Reid did last week, could force some GOP lawmakers facing close 2008 races to either hop on the coming override bandwagon or soften their pro-war stance to please swing voters — or both.

No matter what happens with SCHIP, Democrats vow to keep on provoking vetoes until they find the right issue to overcome the president. There seem to be plenty of opportunities ahead. The White House is threatening vetoes on most of the big bills left on this year’s congressional to-do list, including most of the appropriations measures.

At the moment, the veto threat that appears closest to inciting a successful override drive is against a bill authorizing $23 billion during the next 14 years for more than 900 flood control, navigation and environmental restoration projects— a package that hardly has the same tailor-made public relations appeal for the Democrats as the children’s health measure. But it cleared Congress last month with plenty of override votes to spare, and for precisely the reason Bush has readied his veto pen: On the surface, anyway, it looks like a classic example of runaway pork-barreling.

Still, the bill could afford Democrats the opportunity to test the notion that winning a veto fight with Bush — and the first override since 1997 — would give them momentum at last in the war debate.

Not all Capitol Hill Democrats are on board with the incremental anti-war strategy, provoking a split behind the scenes among top leadership. Pennsylvania’s John P. Murtha said Speaker Nancy Pelosi took him to “the woodshed” last week when he and other House Democrats proposed a war surtax on the very day party leaders were trying to keep the focus on children’s health.

Murtha’s call for an income tax surcharge to finance Iraq’s escalating costs is an example of the sort of direct challenges many anti-war Democrats would rather be aiming at the president. Hoping to appease restive members of their caucus, top House Democratic appropriators said last week that they will not consider another special war spending bill until the White House agrees to a January 2009 goal for ending combat operations in Iraq.

But such direct threats so far haven’t generated enough GOP support to endanger the commander in chief’s firm hold on war policy. That leaves Democrats in the hunt for anything that might weaken a president who shows no concern for his declining popularity and the public’s loss of faith in his Iraq agenda.

The White House appears more than willing to stick to its winning strategy of relying on Republican lawmakers who have just enough votes to blockade the Democratic drive to wind down the war. No matter what the topic of the day might be as the two sides engage in this fall’s series of veto struggles, it’s a safe bet that the future of the Iraq War will hang in the balance unless or until a GOP mutiny surfaces before the president leaves office.

Contributing Editor Craig Crawford is a news analyst for NBC, MSNBC and CNBC. He can be reached at ccrawford@cq.com.