A mx: 49440, recorded New York, NY, June 3, 1918 (take 1); 1918-1922 (take 2).
B mx: 49474; recorded New York, NY, August 5, 1918 (take 1).
Letter code on labels: HY.
(DAHR)
That would be my guess, too, if we were talking about Jazz Records instead of the American Dance Band Discography. Waltzes of any variety were disqualified from inclusion in Jazz Records. There are, however, many distinctly un-jazzy things listed in the ADBD, including multitudinous waltzes and many supposed dance numbers played by military bands. (If it mattered, I'm pretty sure that this track of "Oriental" would be jazzier than many of those. The photo on your video clip strongly suggests that they were aiming for "jazzy", whether they achieved it or not! The "Texas Fox Trot" side rises at least to the level of what they used to call "ragajazz", which is represented abundantly in Jazz Records.)
A more likely scenario is that Rust knew about "Oriental" but for one reason or another did not consider it to be dance music. Perhaps he considered it a concert piece, like the recordings by Victor Herbert's orchestra, which are also omitted from the ADBD. Given the nature of the Fuller orchestra, it would surprise me very much if that were the case, but it could explain the omission.
I think the most likely explanation is simple oversight. The ADBD was a very large project and it was undertaken at a time when the benefits of word-processing, databases and the Internet were either unknown or largely un-developed. Moreover, projects of this type almost never had the kind of editorial guidance or resources they really need. I don't think the ADBD was ever one of Rust's higher priorties, either, as witness the fact that there was never a 2nd edition, while Jazz Records saw 6 editions with constant revisions. I have long suspected that the ADBD resulted from relentless pressure on Rust from his fans and publisher, and that Rust himself probably viewed it as a vexing distraction from the projects he really wanted to work on. Under the circumstances, perhaps it is less surprising that there are omissions than that there are not a lot more of them than there are.
I assume "Oriental" wasn't jazzy enough for Rust's taste, and so he didn't bother including it. The recording date can easily be found, though, in the Discography of American Historical Recordings (DAHR). I've added it above.
In one of those twists that defy explanation, Rust lists side A of this disc in the American Dance Band Discography, but not side B. The listing for side A indicates a recording date of June 3, 1918. The mx shown for side B on the label (49474) would suggest a recording date soon after Side A's. The recording dates and such information as we have on A6000 series recordings in the database suggest a 1918 release date for this disc.