Problem isn't the training, the problem is your back. Heal first Train later and if necessary adjust the training. If you have some belly hanging, then you'll need to reduce that. Lose the belly as it will put stress on your lower spine. You can do this by diet and changing what you eat.
Back injuries take a long time to heal so focus on healing first before starting up full on training. Healing means rest and limiting how much your make your back work. One of the questions you'll need to ask yourself is. "Is my back getting better or worse". You have to monitor your healing. If you are healing fine, then your back pain will become less and then pain will actually go away before the back is completely heal, which is where the biggest risk is. Just because the pain is gone doesn't mean the back is healed. So keep that in mind. Give your back 2 or three months healing time where you aren't training. There's other martial arts stuff that you can can do, that doesn't work your back much. Don't do punching or kicking as this requires you to use a lot of your core. No twisting stuff, but don't start any of this until the Pain is gone.
You will need to stop training 100% as long as you feel any type of pain or discomfort no matter how small, pain lets you know something is wrong. Once the pain goes ease back into training by doing rehab. I agree with Xue Sheng about going to see a physical therapist. Even if it's really just back strain, those guys will be able to give you some nice boring but beneficial exercises for the lower back. They may also have a different perspective of what's going on. Not all doctors have the same knowledge.
I hurt my back about 4 or 5 months ago lifting Gardening Soil. I haven't really felt pain unless I sleep wrong or get out of the car in a bad way. Other than that I feel fine doing everything else. Even though I feel fine, I've restricted myself to rehab exercises. Today is the first day I'll do form training to see how my back reacts to it. I'll do one form and then ease into my training taking note of what I'm feeling. I've also lost some of my belly from changing my diet.
The one thing you don't want to do is train on an injury.