Cut melon in half, cut off peel, and cut fruit into 1 1/2-in. squares, removing seeds, to make 1 qt. Put in a large bowl. Cut heirloom tomatoes into wedges and thick slices. Cut cherry and teardrop tomatoes in half and add all tomatoes to melon. Set aside.
Set lime zest aside. In a bowl, combine lime juice, 2 tbsp. oil, the fish sauce, coarsely ground black pepper, and 1/2 to 1 tsp. Aleppo pepper; set dressing aside.
Clean calamari (even tubes sold as cleaned often need a little attention). First, cut off the pointy tips. Slide a slim knife into each tube. Scrape out squishy bits and the cellophane-like backbone. Rinse well; pat dry.
In a small bowl, gently toss tubes and tentacles with lime zest; remaining 1 tbsp. oil, the finely ground black pepper, and 1/2 to 1 tsp. Aleppo pepper; and the salt. Set aside.
For charcoal, ignite a very full chimney of mesquite hardwood charcoal on firegrate of a charcoal grill. Dump onto firegrate and add half again as much fresh hardwood charcoal; spread level. Set cooking grate in place. Heat 5 minutes with grill uncovered. For gas, heat a gas grill to high (at least 550°).
Meanwhile, put herbs and chile in a small bowl and set aside. Drizzle tomato mixture with dressing and combine gently; set aside.
Working quickly and using long tongs, grill calamari tubes (covered if using gas), turning once, until grill marks appear and calamari is opaque, 2 to 3 minutes total. Transfer to a clean bowl. With tongs, drop tentacles onto grate in several clumps. Grill just to firm up, 15 to 20 seconds; then spread out and cook, turning once, until centers are opaque and tips are browned, 3 to 4 minutes total. Add to bowl with tubes.
Gently toss calamari, herbs, and chile with tomato mixture. Arrange on a large platter.
*Find fruity Aleppo pepper at well-stocked grocery stores and penzeys.com. Find Thai basil at Asian markets and farmers' markets.