I read an excellent conjecture about this in an article by Jason C. Eckhardt from 1987 titled "Behind the Madness: Lovecraft and the Antarctic in 1930", and in a bit of my own research I've found no reason to disbelieve it. Short answer: it seems likely the Dornier Do J "Wal" would have been the expedition's plane of choice.

From a young age, H. P. Lovecraft was extremely interested in science and cutting-edge technology, an interest that was encouraged by his mother and two unmarried aunts. They bought him chemistry equipment and a telescope, and little Howard had access to contemporary scientific literature in the family library and scientific periodicals. By the time Lovecraft was 12, he had written several treatises on Antarctic explorers. His early sonnet, "Antarktos" mentions undiscovered things lurking miles beneath the ice of the continent. As time went on, this only increased, and it is clear from his voluminous correspondence that his interest in scientific endeavor and exploration continued through his whole life.

The story itself was most immediately inspired by the expedition of Rear Admiral Richard E. Byrd to Antarctica 1928-1930, which was widely reported in 1930, including by National Geographic. Lovecraft mentions Byrd and his expedition several times in his letters, including writing about the fossils Byrd's expedition discovered that show that the climate had once been tropical. Details of the expedition match the Miskatonic expedition in the story, including the use of two ships, the stops for supplies along the way, the transporting of airplanes by ship and assembly in the Antarctic, the drilling equipment, the camps, the supply bases, expedition members being trapped by a storm after flying out to do research, etc. On the other hand, Byrd used Ford Tri-Motor planes, not Dorniers.

The airplane you mention is almost certainly inspired by the Dornier Do J "Wal" (Whale), which was used by the expedition of Roald Amundsen, who tried to use two modified seaplane versions of the aircraft to reach the North Pole. They failed in that endeavor, but at that point (1925) the "Wal" broke the record for how far north any aircraft had gone. It was an extremely dramatic attempt, since one of the aircraft was damaged, and the crew had to survive on almost nothing in the frozen Arctic trying to clear a runway for the remaining aircraft to take them all home.

In 1930, Wolfgang von Gronau flew Amundsen's surviving Dornier (another aviator, Irishman Frank Courtney, almost died in it in 1928 after an engine fire left him stranded in the Atlantic) from Germany to New York by way of Iceland and Greenland, and in 1932 he flew another Dornier of the same model around the world. Since AtMoM was written in 1930, Lovecraft wouldn't have known about this latter flight, but it shows that the Dornier was very much in the public eye when it came to cold-weather expeditions. The Dorniers had internal heaters to keep the fuel from freezing, and so perhaps wouldn't have needed the "added fuel-warming and quick-starting devices worked out by Pabodie," but I'm sure there is always room for improvement. Eckhart points out some differences, though, including that the "Wal" had two engines, not one as Lovecraft seems to describe, and that its operational ceiling was roughly half that of the 24,000 feet the planes in the story are said to fly. Clearly, there was certainly a need for these particular Dorniers to be designed especially for "tremendous altitude flying." (Incidentally, the highest peak in Antarctica is roughly 16,000 feet in real life, more believable for the 11,500 feet ceiling of the "Wal.")