Will success spoil the British campaign to recapture control of the Falkland Islands?

That's increasingly the way it looks and sounds. With each grinding and costly military advance, the government of Prime Minister Margaret Thatcher talks more and more confidently of restoring the state of affairs as it existed before the Argentine invasion--and of not much else. That's a definition of quagmire, militarily, in the South Atlantic. It may also turn out to be a definition of big trouble diplomatically abroad and politically at home.

It could be a spoiler for the United States as well--the more so as the Reagan administration, by the imprecision of its repeated professions of support for Britain, nails its flag not to high principle but to particular British objectives, still fuzzy and unrefined.

Already the news is full of analysis of the price American policy will pay all through the Western hemisphere for its "tilt" to Britain. What ought to be seen as U.S. support for Britain--the only available executor of a United Nations resolution demanding Argentine withdrawal--is being interpreted instead in ugly terms: as some sort of collective expression of "Western" neocolonialism, not to say racism. The United States, it is widely agreed, will accordingly be a long time rallying Latin support for its larger East-West concerns in Central America and elsewhere, no matter how the battle for the Falklands evolves. Less obvious is the damage that may well be done to American interests, not only in this hemisphere but in Britain and the Atlantic Alliance, if the battle evolves along the lines sketched out with authority and a certain candor by British Foreign Minister Pym the other day on ABC.

He not only predicts, but profoundly hopes for, a resolution of the conflict by force. At which point Her Majesty's government would take its own sweet time working out with interested third parties or perhaps the United Nations some permanent resolution of the Falklands' fate and that of its inhabitants. "They will want to have a think, won't they," he declared, about "what's best for their children and their grandchildren."

Well, so they will. And so will the British, I suspect, and a lot of other people (including Americans) if things turn out the way Pym was projecting: "It looks as though the Falkland Islands are going to be repossessed by Britain by force-- unless there is a last minute change of mind by Argentina." The foreign secretary emphasized that "nobody wants force to have to be used." But the forceful repossession of the Falklands, he said at another point, was actually "necessary." By doing so "we get back to the position where we were" when the Argentines attacked.

Only, of course, you don't. There is no way to "repossess" the lives lost, the destroyer Sheffield, the frigates, the Harriers, the staggering expenditures, the diplomatic breakage in Europe and Latin America.

Even assuming that the tightening military squeeze on East Falkland Island forces an Argentine surrender and a restoration of Britain's physical control, what will the Thatcher government have accomplished in any permanent sense? That's the first question the British public will ask. The answer will almost certainly take the form of speculation about Britain's ability to maintain its grip on the Falklands and the likely necessity over time to bargain away what it won back on the battlefield.

The second question, as the reckoning sinks in, will be whether this 8,000-mile trip, with all its costs, was necessary. Suffice it to note that over a month ago, when there were as yet no real costs, a Gallup poll found 78 percent of the British public --and 74 percent of Conservatives--of the view that the Thatcher government was justifiably to blame for having been caught off guard. That's a ticking political time bomb that Margaret Thatcher might not survive.

It's always possible, of course, that the British bridgehead will provide a bargaining counterweight that could bring a negotiated settlement short of a British victory by force of arms; Secretary Haig has voiced that hope. But that's not how Pym was talking. And it is not in the political nature of such conflicts. As British costs go up, so must Britain's price for peace; casualties must be redeemed.

The upshot could well be that the United States will find itself less caught up in a worthy crusade for high principles having to do with peaceful resolution of international disputes and more locked into a toughening British position on the merits of the "sovereignty" argument.

This would only confirm the darkest suspicions in the hemisphere, and elsewhere, about the U.S. role. That's why flat-out British military success could be a spoiler all around.