You said yourself, you are over-relying on the muscles of your lower back for stability in your mid section. You have trained yourself to "brace" improperly.
Stand in a door frame. Put your hands on the door frame. Stand on one foot, then raise your other knee up. Like a "high knee." Your spine should stay neutral. Your abdominals and obliques should work with your ql to keep you "straight." If your lower back starts arching and your hips move around, trying to stabilise, BRACE!
Push your abs " out." Keep your back straight. Use the glute on the down leg to really drive your foot into the ground, and "twist" your foot out, screwing it into the floor. Use your abdominals, not your hip flexor to raise your opposite knee.