Cut daikon lengthwise into 1/2-in. slices, then stack and cut into 1/2-in. cubes. Toss with salt in a medium bowl. Let stand about 2 hours, stirring occasionally, to draw out moisture. Rinse daikon well and drain. In bowl, stir sugar with vinegar until dissolved, then stir in daikon, oil, and chiles. Chill, covered, at least 4 hours and as long as 1 week.
Drain daikon, reserving pickling liquid, and discard chiles. In a large bowl, toss together napa cabbage, onions, sesame seeds, and daikon. Add 1/2 cup pickling liquid (or more if you like) and toss to coat. Mix in red cabbage with a quick toss so its color doesn't bleed.