Heat grill to medium (350° to 450°). In a medium baking dish, whisk together soy sauce, oil, three-quarters of garlic, the brown sugar, 1 tbsp. chili powder, the cumin, cayenne, and 1 tbsp. lime juice. Add steak and turn to coat with marinade. Set aside.
Combine mayonnaise, cilantro, green onion, jalapeño, and remaining 1 tsp. chili powder, 1 tbsp. lime juice, and remaining garlic in a bowl. Whisk to blend, then chill.
Oil cooking grate, using tongs and a wad of oiled paper towels. Lift steak from marinade. Grill steak and pepper with lid down, turning as needed, until grill marks appear and steak is done the way you like, 7 to 10 minutes total for medium-rare. Transfer to a platter. Thinly slice pepper.
Spread insides of bread with half the mayo and scatter arugula on bottom. Thinly slice steak across the grain, then arrange over arugula, adding meat juices if you like. Top with pepper, red onion, and bread top. Cut into 4 sandwiches. Serve remaining mayo on the side.
Note: Nutritional analysis is per sandwich using half of mayo.