However, that made me go looking again, in hopes that I might find something useful. And - I might just have done so!
Thanks to the Clarke Historical Library's online archive of Aladdin documents I may have found a possible match. In the 1912 annual catalog, there is a little house called the Sherman. It isn't in the 1910 catalog and there is no 1911 catalog in the archive.
Its footprint is
Here she is in the 1913 annual, with two plan options. Neither is as good a match to my little oddball as the 1912 plan (or to the exterior illustration! look at the windows). The only real difference between this one and the next year's example is that there are people in the image, and the scan is clearer.
By 1915, it's clear that no more changes are being made to this design, or even the catalog page, except perhaps for typesetting. The 1914 and 1915 layouts are almost identical. And, in the 1916 catalog, I think I may have hit upon an answer as to why the Sheridan looks so very very much like my little mystery kit home. In 1916, the Sherman's name is changed to the Sheridan, one plan gets updated, and the exterior image finally gets a full update to match the changed plans.
Look how the unaltered windows (the dining room and kitchen windows have significant alterations in my home) line up with the plan! At this point, while I am very optimistic that this house may, in fact, be a customized 1912 Sherman, because of the differences shown in the overlay image above, it is still equally possible that the 1912 Sherman is a copy of a similar house from a competitor's 1911 or 12 catalog. And, of course, if it is an Aladdin ... why is it all Sears on the inside? What this has given me is a concrete time range for the design of the house. I always had a feeling it was early teens, and now I am certain of it. It makes the search more focused and I am very, very happy about that.