Robear, I just became aware of this post after getting access to archives. I am shocked that your button did not receive the attention it justly deserves. It is a fantastic find!
First of all, absolutely do not dismiss this button simply because it is not listed. For that matter, do not dismiss it because it is backmarked.
Until the "Monroe" button surfaced it was believed that the earliest backmarked political button was the "Jackson" from 1829. The Monroe button pushed the date of that style back another nine years! Examples of "Wellington" commemorative buttons perhaps push the date even farther back another five years.
A backmarked "washington" button recently went up for auction and I believed sold for a a respectable sum. You may want to check "HerItage Auctions", as I seem to remember they sold it along with several others from the Stack auction of Cobb's material. It was somewhat larger than your's, had "washington" impressed on the verso, and had a slightly different and possibly more contemporary eagle device. After comparing your device (eagle facing left, upturned wings, etc. to the earliest use of a similar style device, you'll find that a fairly close example was used as early as the late 1790's on the Albert GI 25B and perhaps a closer match as you get near the War of 1812 period.
The odds may be against his being an inaugural button due to the style, yet the backmarked button recently auctioned (not authenticated, as sometimes it will be impossible to authenticate beyond doubt) which contained the contemporary eagle device similar to Albert 12a etc. makes your's a possibility. Perhaps this button may have memorialized Washington's death, which was around the time of the use of your device. Maybe it commemorated the decade after his death.
One thing is certain though. It is not a maker's mark. It is not a generic political button. Albert took great care in catalogueing known COMMEMORATIVE Washington buttons; even to the point of listing the most humble plastic or glass varieties. He certainly would not have missed this one if it was known, and I would defy anyone to produce examples of more than a handfuls of examples of existing Washington and eagle backmarks. You DO have a rarity, even if it may not necessarily be from the inauguration. Congratulations!
