Reading through the book, I think that the rule about exhaustion is only for days that have a full blown tropical storm. That's the way I read it, anyway. It talks about how on such days the guides know to hunker down; travel on rivers is impossible, and characters who insist on traveling by foot gain 1 level of exhaustion and must save against a second. It then points out that skill checks made to avoid becoming lost are made with disadvantage on storm days.
That's my interpretation of how it is presented in the book. It seems to be in the context of "storm days".
However....given that it's all up to the DM anyway, you can pretty much apply this penalty whenever you want. All it says about determining the weather is that most days in Chult don't go by without some rain, but it may be a mist or a torrential downpour. On days that receive heavy rain, there's a 25% chance of a tropical storm developing.