Beautiful job.
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Try it, the only thing to lose is a few seeds. You might find that good shade will be enough... I grow cool weather plants in the summer in my greenhouse in the shade of the heat loving crops... It gets into the high 30s even 40s in my greenhouse (Celsius) kale, peas, lettuce, Swiss chard, carrots etc do fairly well despite the heat.
I think with lettuce it's the germination that's tricky, I read somewhere you'll run into trouble above 20C (68F if my head is working correctly). But if you can start them in a cooler place, I don't see why it wouldn't be worth a try.