In chapter 16 of Bud, Not Buddy, what are Herman E. Calloway and Miss Thomas fighting about in the kitchen?

Your question is actually an amalgamation of what happens in chapter 15 and chapter 16 in Bud, Not Buddy; however, the answer is rather simple:  the two are fighting about whether Bud should stay with the band. The argument between Miss Thomas and Herman E. Calloway happens first.  By chapter 16, the argument has become a discussion.  First, the reader has to infer what the two are fighting about.  Bud is simply in Calloway’s...

Your question is actually an amalgamation of what happens in chapter 15 and chapter 16 in Bud, Not Buddy; however, the answer is rather simple:  the two are fighting about whether Bud should stay with the band. The argument between Miss Thomas and Herman E. Calloway happens first.  By chapter 16, the argument has become a discussion.  First, the reader has to infer what the two are fighting about.  Bud is simply in Calloway’s daughter’s room and hears the sound of verbal fighting.  It is then that Calloway bursts in and says the following:

I'm going to find out what your game is and … you're going back where you belong.

The reader, then, can safely assume that Miss Thomas was trying to convince Calloway to allow Bud to stay with the band while Calloway vehemently disagreed. 

In chapter 16, the argument has become more of a discussion.  Now the conversation is not just with Miss Thomas and Calloway but with the entire band.  Calloway still insists on sending Bud right back to Flint; however, Miss Thomas is still pulling for Bud:

You have no idea how bad those orphanages can be. … You'll take care of any stray dog wandering through this neighborhood, but when it comes to a child all of a sudden you have no sympathy. … I believe the child ... He's staying right here.

As Calloway still vows that he is going to “find out what the real story is in Flint,” we see Bud getting concerned that he is “eavesdropping” on the band.  In fact, Bud’s eavesdropping is quite important in that it is an interesting way for the author to share information with the reader.  Bud heads quietly back upstairs and “loudly” awakens in order to prevent the band from knowing he heard the whole conversation.