Polar bears would probably survive in the Antarctic, and the Southern Ocean around it, but they could devastate the native wildlife.
In the Arctic polar bears feed mainly on seals, especially young pups born on ice floes or beaches.
Many of the differences in breeding habits between Arctic and Antarctic seals can be interpreted as adaptations to evading predation by bears.
Polar bears would find plenty of fish-eating mammals and birds around Antarctica.
Penguins would be particularly vulnerable because they are flightless and breed on open ground, with larger species taking months to raise a single chick. Bears can only run in short bursts, but they could catch a fat, sassy penguin chick or grab an egg from an incubating parent.
In the Arctic polar bears hunt mainly on the edge of the sea ice, where it is thick enough to support their weight but thin enough for seals to make breathing holes.
The numerous islands off the north coast of Canada, Alaska and north-west Europe provide plenty of suitable habitats. The Antarctic continent is colder, with only a few offshore islands, so bears would probably thrive at lower latitudes in the Southern Ocean than in the Arctic.
We can only hope that nobody ever tries what the questioner suggests. Artificially introduced predators often devastate indigenous wildlife, as it is not accustomed to dealing with them.
This occurred with stoats in New Zealand, foxes and cats in Australia, and rats on many isolated islands.
Large, heavy animals would also trample the slow-growing, mechanically weak plants and lichens of the Antarctic. For instance, Norwegian reindeer have decimated many native plants in South Georgia, an island in the South Atlantic Ocean, since they were introduced 80 years ago.
While, as far as I know, no one has ever been stupid enough to introduce polar bears into the Antarctic, there have been at least two practical attempts to transplant penguins to the Arctic.
The original “penguin” was in fact the late great auk (Pinguinus impennis), once found in vast numbers around northern shores of the Atlantic.
Although no relation to southern hemisphere penguins, it was very similar in appearance, and filled much the same ecological niche as penguins, particularly the king penguins of the subantarctic region.
With any attempt to introduce an alien species, there must actually exist an appropriate ecological niche for it to fill, and it must be vacant.
For the most part, the ecological niches occupied by penguins in the south are filled by the auk family to the north. But the demise of the great auk in the mid-19th century at the hands of hungry whalers created not only a vacancy that one of the larger penguins might neatly slot into, but also a potential economic demand for the penguin’s fatty meat and protein-rich eggs.
It was perhaps the possible economic opportunities that prompted two separate bids to introduce penguins into Norwegian waters in the late 1930s.
The first, by Carl Schoyen of the Norwegian Nature Protection Society, released groups of nine king penguins at Rest, Lofoten, Gjesvaer and Finnmark in October 1936.
Two years later, the National Federation for the Protection of Nature, in an equally bizarre operation, released several macaroni and jackass penguins in the same areas, even though these smaller birds would clearly find themselves competing directly with auks or other native seabirds.
The outcome was unhappy for the experimenters and, most particularly, for the penguins.
Among those whose fate is known, one king was quickly despatched by a local woman who thought it was some kind of demon, while a macaroni died on a fishing line in 1944, although from its condition it had apparently thrived during its six years in alien waters.
And it soon became obvious that the real reason why any attempt to fill the ecological gap left by the great auk was destined to fail was the very reason that the niche was vacant in the first place, such large seabirds could not happily coexist with a large and predatory human population.
Of course, it is the steadily increasing human presence in the far south that is now threatening penguins in their native habitat.