It turns out that this engraving is based on a photo (or is it a drawing?) that shows up on this poster for the 1895 season - however the photo negative was loaded backwards (note the "S"s on Sousa's collar):
So here is that photo flipped and enlarged, and - voila! - it matches the engraving on the bell (note in particular the embroidery on the jacket):
Now, when was this photo taken? We can't say for sure, but based on extant photos of Sousa from 1892 and 1893 (see below), it would seem that this particular uniform was new either in 1894 or 1895. And, of course, the Sousaphone had to be made after the photo was taken, eliminating 1893 as an option (well, unless the horn originally didn't have an engraving on the bell, but it was added a year or more later).
1893 (month not specified)