Yes it is a real first flight cover. Aerodacious and that
ebay seller are using the old, outdated American Air Mail Catalogs and not the new American Airmail Catalogs. This cover is listed in the AAMC Seventh Edition, Volume Three as Airmail Route 20 which replaced the old Contract Air Mail Route 29. There were five legs on this inauguration by the US Post Office's contractor, Robertson Airplane Services Company, between New Orleans and Beaumont so one would need to collect five covers to would make the full set. The five, with their proper catalog numbers are AM20W1 from New Orleans and flying westward; AM20W2 from Baton Rouge fying westward; AM20W3 from Lake Charles flying westward; AM20E3 from Lake Charles flying eastward; and AM20E4 from Beaumont flying eastward. At this point in time most but not all first flights were getting backstamped. This is one of the examples where backstamps were not being applied. Your particular cover is an AM20E3 flying eastward from Lake Charles. You can tell by the postmark which shows a time of 10:30 AM. The westbound covers - AM20W3 - have a time of 2 PM in their postmark. Your AM20E3 is the most common of the five legs of this inauguration. There were 27 pounds of mail (about 1,000 covers) on this leg which seems like just a few but in terms of firrst flight covers this is a substantial number. So the
ebay seller is just over-hyping their cover by calling it "Rare". The Catalog gives is a value of $5 which would be a full retail price - otherwise it would normally be found in the $1 to $3 range.