I don't dispute any of the points you make, xiphophilos, but offer some others to consider:
1. Columbia labels continued to cite patent nr. RE. 16588 and 1702564 at bottom through 1938 (label style VIII.B.2 in M. Sherman & K. Nauck, Note the Notes, page 33).
2. Although Otto Heineman sold his General Phonograph Corp. to Columbia in the fall of 1926, OKeh labels didn't cite the Columbia Phonograph Company, Inc. until much later. I think the Columbia notation corresponds roughly to when Grigsby-Grunow purchased Columbia in December 1931. I've been trying to determine what the latest new release to cite the OKeh Phonograph Corporation was, because this would provide a clue to the date of this record. I will say that this Louis Armstrong record is listed with the OKeh Phonograph Corporation credit in Discogs, and that is the original 1931 pressing, I believe.
I think that date is a bit too late. I don't believe you have a repress at all, but this is the original release from 1931.
This Okeh label corresponds to Columbia's 1930-1932 Viva-tonal style label marked as "Viva-tonal Recording" with patent nr. RE. 16588 and 1702564 at bottom.