"BG On The Record" provides a little more information about this controversy. As has been previously stated, Benny Goodman insisted that "Clarinetitis" was made in Chicago in February 1927, while the Brunswick recording ledgers list the recording date as June 13, 1928, and the recordings bear both a Chicago (C-prefix) and New York (E-prefix) matrix number.
According to "BG on the Record," Benny was in Chicago in the winter of 1927, and in New York in the summer of 1928, playing with the Ben Pollack band. But in New York, Pollack's luck ran out, and his men were forced to find side jobs in the recording studios and with other bands in order to keep eating. Jack Pettis recorded three tunes on June 20, 1928 in New York, and for some time collectors insisted that Benny was the clarinetist on those sides. But Benny flatly denied that he was on these particular Pettis records (although he did play on other Pettis dates) and subsequent research has indicated that the clarinetist is indeed Don Murray.
The E-prefix matrix numbers for the Pettis sides are consistent with the E-prefix matrix numbers assigned to Benny Goodman's "Clarinetitis" and "That's A Plenty" (thus dating both sessions to June 1928) but "Clarinetitis" and "That's A Plenty" both also have C-prefix matrix numbers, indicating that they were recorded in Chicago. We know that Benny briefly traveled with the Sam Lanin band to Chicago in the summer of 1928 for a short engagement at the Congress Hotel, but the exact date of that trip is unknown.
To me, it seems most likely that "Clarinetitis" and "That's A Plenty" were recorded in Chicago in February 1927, yet for some reason these two sides were re-logged in June 1928 with New York matrices. Perhaps the sides were readied for issue in June 1928, but the paperwork from the Chicago studios was lost, and so the records were re-logged in NYC.
Does anyone know when the "first pressing" of "Clarinetitis" hit the streets? That might help to shed more light on this dilemma.