Wait, what? I haven't found heat to be any kind of issue for real Ameraucanas. I'm in Florida and they do great here.
The only drawback is waiting a bit longer for laying onset, they're right around 5-6 months.
Of the Ameraucana we've had in the past, we had one everyday layer who never rested (a Silkied splash) and many who lay 4-5/wk. So they're not a production breed but they're not slouches either. Large eggs are normal for the breed (from breeders).
A lot of laying traits depend on the line, that's especially true of Ams which have been shown a lot (black Ams tend to get all the highest titles) so breeders have hoarded the best blood. And then you have hatcheries making a mess with the name. So sourcing good birds is a big headache.
For the Op, WTB may fit the production characteristics better. I'm personally a little too picky about looks for the surprise color factor, but some people love that.
Another factor to consider is temperament. Real Ams have very mild mannered girls who tend to get picked on by tougher breeds. But they are also clever, alert birds who do well ranging / free ranging.