I have to recognize that I’m quite limited in my ability to give helpful feedback through this kind of forum because of all of the complexities and variables that exist within each individual’s circumstance. If I had the benefit of a thorough evaluation, it would be much easier to give you specific help based on your needs. I will do my best to apply some general principles that may help.
Firstly, great work in your drive to help yourself. It’s frustrating that you have experienced this setback with your back pain. It is likely that fatigue of your extensors is playing a strong part in your symptoms, but it may not be as simple as getting them strong. It is possible that your extensors are overactive due to excessive strain where continued weight loss will be very helpful. In the short term, if your hips and trunk are more stable (gluteal and abdominal strength), and if your back isn’t fighting against stiffness in your hips and thoracic spine, your extensors may be able to relax and function more normally.
Quite often there are strong elements of excessive and unconscious guarding of the extensors. We see this with people who have fear and anxiety built up around their back, and through no fault of their own, their back muscles remain locked up. Such people often hold their breath when they move, and they don’t trust their spine. Many of them have been given information on degeneration, and arthritis that makes them feel their spine is fragile. This can be the key thing that drives someone’s pain.
In your circumstance, it is very likely a combination of these factors. My best advice is to have a good physical therapist assess and understand all of these variables and map out a road to recovery for you. Slowly working on progressing your standing and walking endurance is a good idea. Not pushing past your endurance too much, but challenging yourself is a good premise. It sounds to me like it is more complex than gaining endurance with your back muscles. Like I touched on before, if your hip mobility and thoracic mobility (particularly the ability to stand up straight) are limited, your hips and pelvis rock and tip a lot while walking, and your muscles don’t control the position of your spine very well, this will cause a lot of load on your back muscles.
Your pain is unlikely to be a warning of significant damage in your spine. The symptoms you feel don’t sound like classic DOMS, as you don’t have normal muscle activity and tone there.
To understand the interplay between all these variables and to have a staged program of exercise and intervention, you will likely need some expert help. I would expect any such program would include continued weight management, gluteal strengthening, developing control of pelvic ‘tilt’, and perhaps some manual therapy to help unlock these muscles.