Make pickles: Whisk vinegars, sugar, and salt in a bowl with 3 cups water until sugar dissolves. Halve French radishes. Slice kohlrabi, beets, daikon, and black radish into very thin (1/16-in.) slices--easiest using a mandoline or food processor with slicing blade. Cut watermelon into 1-in. dice. Put each vegetable or fruit in its own jar, gently tamp it down, and cover with brine. Chill, covered, 1 day.
Marinate beef: Whisk soy, sugar, corn syrup, rice vinegar, garlic, ginger, sesame oil, and pepper in a bowl with 1/2 cup water. Put beef in a 1-gal. resealable bag, add marinade, and chill overnight, turning a couple of times.
Heat grill to medium-high (450°). Oil cooking grate with a wad of oiled paper towels. Lift steak from marinade (throw away marinade). Grill steak with lid down, turning as needed, until grill marks appear, 7 to 10 minutes total for medium-rare. Transfer to a platter and let rest 10 minutes.
*Black radishes have snowy white interiors and a spicy kick. Find them at farmers' markets and some grocery stores.
Make-ahead & packing tips: Make pickles and marinate beef 1 day ahead. Bring pickles to beach in jars and beef in its bag, all kept chilled.
Note: Nutritional analysis is per serving.