Did you catch the obvious?
Yes, you are correct. Its not a 1922 Hayes FDC. But it is a Hayes FDC. Improvements in stamp producing technology led to replacing flat printing presses with more economical rotary printing presses. The rotary press Hayes stamp (Scott 692) was released on September 4, 1931 in Washington, DC.
Besides the date, there are other clues about the above cover as well. The 2-cent Washington stamp (Scott 554) was not released until January 15, 1923. Remember that the Hayes stamp was the first of the new series of stamps. The Hayes color in 1922 was Peacock Blue. By 1931, the color was more of a blue green. The air mail franking is a less obvious clue. In 1922, the air mail fee, between selected cities, was 6 cents per oz. Of course the 2 cent first class mail rate would have been appropriate for delivery from Washington to nearby Cherrydale. Actually, the 15 cent franking for air mail is an overpayment since the rate in 1931 was 10 cents per 1/2 oz. The handwriting of servicer Henry Hammelman would not be a clue since he did make 1922 Hayes Washington FDCs.