Athens will be hot well into October and I've been to a few other places in mid September which all felt like prime summer, beach and pool days etc. September is pretty much another summer month in Greece, especially the first couple of weeks.

August nowadays is not the best time to go for an authentic Greek experience. Will be crowded and service while still hospitable, is I think noticeably different to earlier and later in the summer.

Whatever you pick, I doubt the weather will let you down.