Put the potatoes in a medium pot and cover with water by about 2 inches. Add a pinch of salt, cover and bring to a simmer over high heat. Reduce the heat to medium high and simmer until just tender, about 5 minutes. Add the snap peas and continue simmering, covered, until the peas are crisp-tender, about 3 more minutes. Drain the potatoes and peas and rinse under cold water; transfer to a large bowl.
Make the dressing: Puree the tomatoes, olive oil, mayonnaise, vinegar and 1 tablespoon water in a blender.
Divide the lettuce among 4 bowls and drizzle with some of the dressing. Add the tuna, bell pepper, olives and basil to the potatoes and peas; add the remaining dressing and toss gently to coat. Season with salt and pepper, then pile on top of the lettuce.