The Smithsonian Air and Space Museum blog has a great article about Wolfgang von Gronau's around the world journeys using the Dornier Do-J Wal. It includes some gorgeous high resolution pictures of the D-2053, a specially modified Wal, taken during his 1932 circumnavigation.
The only significant difference between this aircraft and the ones in Lovecraft's tale would be the upgrades installed by Frank Pabodie. Conjecturally, that would include an enclosed cockpit, the use of aluminum to replace some of the plane's steel construction, and slightly lengthened wings. That's the only way to get the plane to match the levels of performance described in the story, at least from a Watsonian perspective.
This is one of the very few overhead shots of the Wal I've ever seen. As the scoring on the tail rudder demonstrates, "clean burning" was a pretty relative concept for early mid-century aircraft engines.
Gronau's earlier excursions used the D-1422, the exact same plane flown by Roald Amundsen (wearing the N-25 badge) during his 1925 arctic expedition. That specific plane is almost certainly the one that inspired the aircraft flown during the Miskatonic Antarctic Expedition in "At the Mountains of Madness". Amundsen's expedition had cemented it's suitability for polar exploration. On August 26th, 1930 Gronau and his crew completed an epic flight across the Atlantic with the D-1422 and landed in the Hudson at New York City. For the rest of the year von Gonau made regular appearances in the news as he and his crew were feted at events across the United States and Europe. Lovecraft penned ATMOM in February and March of 1931, and it's no great stretch to see Gronau's flight influencing Lovecraft's portrayal of the aerial assets in the Miskatonic expedition.