I assume you are referring to a combined visit of either the north or south Pantanal and the Amazonian birding found around Alta Floresta, using Cuiabá as a sort of transportation hub.
September is widely regarded as the best month for this trip. There is little rain, but at least it has started by this time, enough to get a number of species singing and thinking about breeding. The rains also put out a few of the fires which are commonplace (though apparently this is improving over the past couple of years) starting in August, so smoke shouldn't be a big problem. The water levels in the Pantanal are at their lowest in early (?) September, so this would be the best time for the wildlife spectacle for which the area is famous.
If avoiding rain is you #1 criterium, try a visit in June or July. The birds aren't particularly vocal, but antswarms are active in this period, and you certainly won't get any rain. Insects don't pose a big problem around Alta Floresta/Cristalino - there are far fewer mosquitoes in the dry season than say, January or February when it can be tough to breathe without inhaling the rascals. Though I have only visited once, I get the impression the mosquitoes are difficult to avoid in the Pantanal.
Here is a rough outline of rain periods in Alta Floresta (note that this is really only helpful for Alta Floresta and Cristalino only, as these rain patterns vary throughout the Amazon basin):
January - daily rains; it might rain all day on an average of once per week
February - daily rains; it might rain all day on an average of once every ten days
March - almost daily rains; highest water levels of the year
April - frequent rains, but no longer daily; water levels begin falling in early April, dropping rapidly by the third week
May - occasional rains, pleasant temperatures; water levels still reasonably high, but all trails open
June - no rain; plenty of sun, high temperatures but comparatively low humidity; low vocal activity
July - no rain; plenty of sun, high temperatures
August - occasional night rains towards the end of the month; possibly smoky from forest fires; a number of fruits ripening and seeds dropping, good for frugivores
September - occasional rains, at night or in the afternoon; vocal activity picking up
October - overnight rains refresh the forest; vocal activity better
November - rain in the afternoons and overnight, humidity very high; decent vocal activity
December - rains intensify, high humidity; many birds still vocal and breeding
Please keep in mind that the above is a rough estimate of precipitation and bird activity; the onset and duration of the rainy season varies from year to year, but generally speaking runs from December to April.
hope this helps a little,
Brad
