I have a "starters" variety of laying chickens.

Barred Rocks, RIRs and EEs.
They Eggers don't really like laying in the Summer heat, but have always laid massive eggs when they get the chance!
The Barred Rocks seem to do the best for me, they haven't had a single health problem and are very easy to find.
My RIRs get sick from the heat sometimes and I have to take them inside and give them a saline drip... It's rough.
I also have a few bantams and exotics (Silkies and Sultans).
Obviously the exotics don't do so well in the normal coop, so I just built them a seperate short pen, essentially a run with shade, in my garden where it is VERY cool and they get along just fine! They don't kill my plants or anything like that so it's great! c:
The Bantams do extremely well in the heat! Maybe it's their size or something? But they lay teeny tiny eggs. Maybe you could just get a whole bunch of them to compensate for the small eggs.

Also, we have a lot of bantams here in rescues because they're used in fights and are bred like crazy for that.
The most important thing here is to get a mister and a dust bath, maybe even an auto waterer!
My mister is just a strip at the top of my coop, not too close to the bottom because I don't want the floor to get mucky and humid. And my Dust bath is a small pool I got at ace for $10, filled with gravel that someone what giving away at the curb. I don't have an autowaterer, but I'm a crazy bird lady, so I go refill their water every evening.
Salvage is your best friend here by the way! If anything I said seems like it's be too expensive (ex. 2nd bantam coop) then go curb hunting and you're sure to find some good stuff!