Hi Debs, I'm hoping to do this amazing tour in Sept next year and I've done a bit of research on climate, which may of course be contradicted by folks who have actually been there at those times! I have friends who went at the end of May this year and it was very smoggy in the cities and hot/very humid as well, so I'd say June would be similar and probably a good bit hotter too. My preference would be to go in Oct, it would be cooler and possibly some rain but I think it would be better than high heat/smog/humidity etc, but that's just my take on it! Also with climate change, many parts of the world are getting unseasonable weather which can be colder and wetter than expected, as well as hotter than the norm! I went to South America last Sept and it should have been +20C in Chile and it was barely 10C!! The only bit of advice I'd give you is that after reading many, many threads on this tour and looking at Trip Advisor (which I do take with a pinch of salt), it really does seem worthwhile to upgrade on the cruise - I certainly will be!
I hope you pick the right date to suit you best, if you choose Oct and I get to go in Sept, I'll put up a review as soon as I get back!
Hope this helps in some way...
Bob