I'll add my 2 cents. Sometimes the lower back pain can be not just from the added weight, but from how you handle the weight. You may be careful about keeping your core tight and keeping the natural lordosis (curve) in your lower back while you are lifting, but might be picking up the bar or putting it down in a way that can strain your back. When picking the bar up from the ground, try putting one foot slightly in front of the other, then bending with your legs. Push up from your legs when lifting the bar from the floor, and make sure to keep your core strong (no rounding of the back---the most risky position).
Do you have any ankle weights that can be fastened together to make a weighted "belt"? If so, you might try those for the final 10 pounds you want to lift. The weight would then be over your legs, and put resistance on the lower body, but would not put too much weight on the lower back.
You're probably right in that your core strength has a lot to do with it, so keep up those planks!