You are right that autoflower plants should NOT BE transplanted too much. However your climate /temperature is not optimal, so some shade is probably called for. I have just started playing around with autoflowers myself and I am growing indoors with them for the first time. Container size is about right though. You might try evaporative cooling, using cheese cloth as both shade and a way to cool the plant some. Dampen the cheese cloth during the hottest part of the day. Just keep the cloth from touching your plant. Cheese cloth is the cheapest thing that I can think of to help you out.
I still start my autoflower seeds in 4" pot and then transplanted into the final home container. I have had no problems with this method so far. It's just easier for me that way.