This post may sound like  a throw back to the  aerobic era of the 80’s and 90’s, but based on over 15 years of clinical training experience, I’d say its an error to factor out of your training an effective aerobic component.
It should always be there.  In the issue of bad backs, its essential that you commit to regular physical activity,  preferably outside.  At the very least, if you are walking around with a locked down core,  a natural curve in your back, you are not sitting slumped!
This does not mean hours of Long Slow Distance, but it certainly means 20- 30 minutes of low to moderate  back sparing “cardio” every day. This could be cycling to work, a fast lunchtime walk, or a jog or a run. There are lots of  benefits to underpinning cardio activity, but, when you are suffering from back pain, a few natural endorphins  pumping around won’t harm you