My lettuce is growing in a salad table in rich potting mix, with dappled sunlight behind the giant tomato vine and at the outer edge of the oak tree. Lettuce is very happy in this setting.
I can’t get it to germinate, but the place I have the seeds is similar to your description. Maybe the dirt isn’t rich enough or maybe lettuce can go in boxes. Arugula, on the other hand is growing like gang busters in the row over from the lettuce. I didn’t even thin the plants. It just keeps giving.