In a jug, combine the lime juice, palm sugar, fish sauce, sesame oil, soy sauce, ginger, and crushed garlic. Whisk until the sugar dissolves and the mixture is well blended. Place the beef rump steak in a glass or ceramic dish and drizzle half the dressing over the meat. Cover the dish with plastic wrap and refrigerate for 2 hours, turning the steak occasionally to ensure the flavors penetrate evenly.
Preheat a barbecue grill or a chargrill pan to high heat. Cook the marinated steak on the hot grill for 2–3 minutes per side for medium, or adjust the cooking time to achieve your preferred doneness. Once cooked, transfer the steak to a plate, cover it with foil, and let it rest for 10 minutes to allow the juices to redistribute.
In a large bowl, combine the grape tomatoes, cucumber, onion, red chillies, mint, coriander, Thai basil, peanuts, and shredded makrut lime leaves. Thinly slice the rested steak against the grain to ensure tenderness and add it to the salad. Drizzle the remaining dressing over the ingredients and gently toss until everything is evenly coated.
Divide the salad into individual bowls and serve immediately for a vibrant, flavorful dish.