You know who was using bass saxes in the 30s? School bands. Almost exclusively.
I wonder, however, if there weren't a number of "hotel bands", playing the music of 15 years earlier, that would have been using bass sax too. You know, a 32 year old businessman and his wife might well want to go downtown to the Baker Hotel and dance to the music that was popular when they were 20 years old; i.e., covers of old Bix tunes and so on. In digging around about the life of Adrian Rollini it looks like his last recording on bass sax was around '36 or so. By 1936 the vanguard of jazz as preferred by aficionados and the young people who make trends had long since left the bass sax behind, but I bet there were a lot of older people still digging the older sounds and bands supplying it to them. This is the "underbelly" of popular music we don't hear much about, just like all the bands of 50 and 60 somethings today playing 3 chord rock and roll from the sixties and seventies in bars to 50 and 60 somethings. They're not doing anything new, but they're still out there.
Think how long Guy Lumbago, Les Brown, the Elgarts, and their ilk kept going, playing a repertoire largely from the mid 20s through the mid 40s, well into the 1990s.