My guess is that your core is weak, and the cycling is showing it up. Do core exercises twice a week, see if that helps. Situps, back extensions, leg lifts, Roman chair, barbell squats, deadlifts, good mornings, all that stuff. Start easy and gradually increase over a period of weeks. Biking demands a really strong core. One wouldn't think so, you're just sitting there, right? But your core is the base of all movement.

Oh . . . and a back brace is exactly the wrong thing.