This does seem like a puzzle. Patients being treated for cancer who are so tired ask the same question. However, exercise has been proven to work for both problems: back pain and fatigue! In fact, when it comes to back pain, there's no evidence that the back pain gets worse with exercise. It frequently gets better!
Exercise should begin slowly. It may be best for you to start on a day when you are having some pain so you won't overdo. There are three exercise goals for patients with chronic back pain of unknown cause. The first is to improve flexibility and strength. The second is to reduce how strong the pain is (the intensity). The third is to increase function even if pain doesn't go away. In other words the patient will be able to do more even though it still hurts.
Many, many studies have shown exercise is the key to overcoming back pain. If you need help getting started, see a physical therapist. You can get set up on a home program with monitoring as often as you need it.