Seems the story is just a legend.
The first attempted murder on the continent occurred in 2018, according to a NY Post article from October 30, 2018:
Antarctica scientist stabbed colleague for spoiling book endings
By Natalie O'Neill October 30, 2018 2:30pm
In the first attempted murder ever on the frozen continent of Antarctica, a Russian scientist reportedly snapped and allegedly tried to stab a colleague to death because the victim kept giving away the endings of books.
The earliest mention of the chess game "murder" seems to be a 1985 piece in the Wall Street Journal, which that Carl source from 1988 you linked cites as its source for the story.
That piece, "Polar Privation: Antarctic Life Proves Hard Even for Those Who Love Their Work" by WSJ staffer Bryan Burrough (WSJ December 10, 1985, p1) offers nameless "NASA psychologists" as the source:
The Soviets, who maintain seven scientific stations here, have already applied one lesson: According to NASA psychologists, cosmonauts have been forbidden to play chess in space ever since a Russian in the Antarctic murdered a colleague with an ax after losing a game of chess.
(That paragraph is everything the WSJ article has to say of the incident)
Tellingly, an earlier story from the New York Times about the stress of staying over a winter in Antartica identifies as a "milestone in the history of the South Pole" a drunken brawl that occurred in 1979, painting it as the worst psychological breakdown there to that time, as if no earlier murder had happened.
Strife and Despair at South Pole Illuminate Psychology of Isolation
By Robert Reinhold
Jan. 12, 1982
SOUTH POLE STATION, Antarctica
The events of Aug. 17, 1979, will not get into the official history books, but they were something of psychological milestone in the history of the South Pole. Outside, temperatures dipped to 71 degrees below zero in a blinding blizzard, but inside, emotional temperatures among the crew of 16 men and one woman, after nearly half a year of total darkness, were reaching the flash point.
Foaming at the mouth and roaring drunk, a member of the crew who had recently learned of his father's death, piles into the galley in rage. He yells and begins to smash cups wildly. Blood and glass everywhere. Soon he spies his rival for the affections of the station's lone woman and charges with a two-by-four, then runs out into the blizzard. It is hours before the mayhem ends, with gashes, bruises and frostbite.
Certainly some the "believable details" given to help prop up the murder tale are spurious, namely that the murder (or attempted murder) was a factor in establishing an international law regime there and that chess has been banned either there or in space.
The first conference to establish a law regime in Antartica was organized in 1958 (invitations officially sent to the various countries on May 3, 1958). The impetus for this activity was not a chess game or an attempted murder or a murder, but rather the simple presence of the Vostok base and the Mirnyy base in the Australian Sector.
From The Soviet Union and the Antarctic Regime, Boleslaw A. Boczek
, The American Journal of International Law , Oct., 1984, Vol. 78, No. 4 (Oct., 1984), pp. 838,839:
The presence of the Soviets in the Antarctic nevertheless produced mixed feelings in the Western capitals. The Australians, in particular, in whose sector the first Soviet stations were established, were concerned about the military potential of these stations if the Soviets had any hostile designs.
It was in this atmosphere of hope and anxiety that the United States convened a conference to negotiate a legal regime for Antarctica. The following bases were proposed for the regime: that there be free access by all to carry out scientific investigations; that Antarctica be used for peaceful purposes only; that Antarctica be a nonmilitarized area for which a right of inspection would obtain; that the legal status of the claims be frozen; and that an administrative arrangement be set up to manage the projected regime. Notes containing the invitations were delivered on May 3, 1958 to the governments of the 11 other countries participating in the IGY Antarctic projects, including the Soviet Union.25 In accepting the invitation, the latter expressed general agreement with the principles of the U.S. proposal but reiterated the reservations concerning the territorial claims that it had stated in its Memorandum of 1950.26 Despite some obstruction at the 60 preparatory meetings,27 the conference opened on October 15, 1959, and the Antarctic Treaty was signed six weeks later on December 1, 1959.
The 1959 bit matches the year of the supposed chess game and murder, but there's no mention of any murder or any other person-to-person crime there.
There's also no mention of anything so nefarious in Antarctica Ahoy!: The Ice Book, which journals the Third SAE (which operated from the base from 1957-1959). But the author's clear bias may make the omission somewhat less of a compelling argument. https://en.wikipedia.org/wiki/3rd_Soviet_Antarctic_Expedition
Also, the assertion that chess was banned in the Antarctic seems false, as evidenced by the tournament held there in 1978:
Back in 1978, Soviet polar explorers, exploring the ice continent, held their first team chess championship. As a result of a five-month "battle on the ice", the team of the station "Novolazarevskaya", headed by a candidate for master Yury Evdotiev, won. The following places were taken by "Bellingshausen", "Molodezhnaya", "Mirny", "Vostok", "Leningradskaya".
Nor did the Soviets ban chess in space. For instance:
In June 1970 the Soviet astronauts Andrian Nikolayev (Mission Commander) and Vitaly Sevastyanov (Flight Engineer), who were aboard the Soyuz 9 spacecraft, played a consultation game against the earthbound Viktor Gorbatko (also an astronaut) and Nikolai Kamanin (Soviet general, and head of the astronaut training program).
In any event, more directly to your inquiry: none of the retellings of this tale ever even mention the victim or the attacker's name, much less the details of the chess game, so no, there are no details about the chess game itself (other than "it wasn't a draw").