So, the short answer to this is that no, one French privateer running amok in the Pacific will not tip the balance of the war in Napoleon's favor, in whatever year or hypothetical year (more on this later) that the movie was set in.

(I believe the opening credits of the movie name a year, but I don't remember what it is and I'm sadly away from my copy of it. But it's a moot point here.)

The slightly longer answer is that Master and Commander: The Far Side of the World is a movie that's based on The Far Side of the World, which is the tenth of a series of naval novels written by Patrick O'Brian.

In the book, HMS Surprise is given a mission to capture the USS Norfolk, a frigate in commission that has been sent into the Pacific to attack British whalers. For the story, O'Brian drew inspiration from the actual capture of the real-life USS Essex by HMS Phoebe in 1814, though because he set the first of his series of novels in 1801 and did not anticipate writing 20 of them at the time, he started to play a bit fast and loose with chronology. To quote from the preface of TFSOTW:

Perhaps few authors are wholly original as far as their plots are concerned; indeed Shakespeare seems to have invented almost nothing, while Chaucer borrowed from both the living and the dead. And to come down to a somewhat different plane, the present writer is even more derivative, since for these books he has in general kept most doggedly to recorded actions, nourishing his fancy with log-books, dispatches, letters, memoirs, and contemporary reports. But general appropriation is not quite the same thing as downright plagiary, and in passing it must be confessed that the description of a storm's first aspect on p. 308 is taken straight from William Hickey, whose words did not seem capable of improvement.

If these tales are to continue, however, it is clear that the writer will soon have originality thrust upon him, for he is running short of history. Some ten or eleven years ago a respectable American publisher suggested that he should write a book about the Royal Navy of Nelson's time; he was happy to agree, since both the period and the subject were congenial, and he quickly produced the first of this series, a novel based upon Lord Cochrane's early days in command of the Speedy, which provided him with one of the most spectacular single-ship actions of the war as well as a mass of authentic detail. But had the writer known how much pleasure he was to take in this kind of writing, and how many books were to follow the first, he would certainly have started the sequence much earlier. For the 14-gun Speedy did not capture the 32-gun Gamo until 1801 and this was followed by the ill-judged Peace of Amiens, which left enterprising sailors less time to distinguish themselves than they could have wished and deprived later writers of a great deal of raw material. Historical time has not yet run out for these tales, and in the present book the naval historian will detect an echo of HMS Phoebe's pursuit of the USN Essex; but even in the early nineteenth century the year contained only twelve months, and it is possible that in the near future the author (if his readers will bear with him) may be led to make use of hypothetical years, rather like those hypothetical moons used in the calculation of Easter: an 1812a as it were or even an 1812b.

Yet if he should do so it will be strict chronology alone that is affected; he will continue to respect historical accuracy and speak of the Royal Navy as it was, making use of contemporary documents: the reader will meet no basilisks that destroy with their eyes, no Hottentots without religion, polity, or articulate language, no Chinese perfectly polite, and completely skilled in all sciences, no wholly virtuous, ever victorious or necessarily immortal heroes; and should any crocodiles appear, he undertakes that they shall devour their prey without tears.

So the obvious point of comparison here is that the book is recounting the exploits of the Essex during the actual War of 1812, specifically the pursuit of Essex by Phoebe and Essex's capture in 1814. The movie, being released in 2003 during the era of Freedom Fries, among other things, changes the antagonist from American to French, which alters the context somewhat. There's nothing in particular to think that a single French privateer would have done more than dent the whaling trade in the Pacific, causing unease among those invested the whale fishery but hardly ending the war. The movie is building tension with that statement, along with Aubrey's "exceeded my orders long ago."

In the book, Aubrey's mission is to chase down the Norfolk and he's immediately ordered to take in six months' supplies, as he could possibly intercept it off the coast of Brazil but may well have to chase her around the Horn. The book differs from the movie in several other aspects -- not to spoil too much of it, but the Surprise is crippled off the coast of Brazil by a storm that destroys her bowsprit, not by an ambush; the Jonah Mr. Hollom does not commit suicide, but another crew member does; Maturin's role as a spy is never mentioned; etc. None of that is a knock on the adaptation, which is still my favorite movie depiction of the Age of Sail, but things were necessarily simplified due to the nature of film vs. literature.

In the actual War of 1812, the US Navy had some notable success in single-ship actions, with Constitution fighting and capturing the frigates Guerrière and Java and the United States taking Macedonian; the British won the fight between HMS Shannon and USS Chesapeake, but all of these were nuisances to the British government rather than anything that would come close to being decisive. American privateers did make inroads in to British shipping, briefly driving Lloyd's insurance rates for West Indies shipping up close to 30 percent, though overall rates did not change in this period from those in 1810 or 1811. By 1813, the British navy had been able to enforce a blockade of American ports, and privateers from Canada had made major inroads into American shipping before blockade. The war, which was ill-conceived to start with, was extremely unpopular in New England, and Quakers there enforced neutrality at their ports, meaning the British could buy supplies locally. The Essex was actually the last US frigate at sea, and captured a handful of whalers before being hunted down by Phoebe; the President attempted to escape New York in 1815 and was crippled by HMS Endymion, and captured within a few hours.

Regarding this, though:

Surely they could launch a sortie from India to intercept this lone unaided French ship?

The issue is that there wouldn't be any way to inform people in India that the ship was in the Pacific without sending a ship to India; at the point in which Surprise is chasing Acheron around the Horn, it would be quickest for it to serve as its own messenger.