One legged presses. Do hack squats cause you a problem??? Stick with one legged anything because that takes half the weight off of your back but your legs won't know the difference. Even squats just put one leg behind you up on a bench and squat like normal but you'll only be doing one leg at a time. Takes a little to get the balance down but it's better then having chronic back problems.
Get rid of the belt it's hurting you more than helping. Work on your core, planks whatever. Try a plank on your elbows, if you can't hold it very long that would be a major problem to work on. If you're shaking and falling apart at 10 seconds then you have a weak core for sure. Strong core = strong back.