Most situations dont allow the chiller to be vented outside at all. If you have some way to vent the chillers hot air "exhaust" into the chimey WITHOUT restricting the airflow that blows out the back of the chiller, then up the chimney is better than in the house. My chiller is in the fish room and vents out into the living room. It is along the front wall and we do not notice a heat increase at all when it is running, even in the summer. It may be because we have cathedral ceilings and the hot air rises, probably before we can feel it since the chiller vents a dozen feet away from the actual sitting area.
One problem though ... the warmer the ambient temp. around the chiller the less efficient it cools. 95 degrees ambient is a common maximum temp. that some chillers mention. But I can tell you that cooler is much better, makes a big difference in run time for the chiller.
Im not sure about your fan cooling situation. Do you only use a fan in the canopy? Or do you also use fans to blow across the surface of the water in the tank and sump? Yes, replacing lots of evaporated water can be another issue that has to be addressed but it also makes the chillers job easier. If humidity is that much of an issue you may very well have to ease up on the evaporative cooling and let the chiller do more of the work but the air the chiller blows out is also humid because it is generally exhausting air that was drawn in from a humid environment.
But if you do deacrese evaporative cooling and the humidity is decreased because of it then obviously the air out the chiller exhaust is less humid as well. If you locate the chiller remotely, obviously the situation changes.
One more thing ... I would upsize the chiller if it is going to be working in a 90 degree environment.
I never had a basement, isnt it more humid in the basement in the first place because they are basically underground?