While on a Keto Diet, do not assume that it's best to simply give up on exercise. It would remain an important part of maintaining your health. However, what you could easily do is, include low-intensity exercises such as walking, light jogging, or even gentle yoga for about 30-45 minutes daily. The idea is to maintain a steady-state of physical activity and avoid workouts that demand a sudden burst of energy.
Some diets are highly inflammatory, especially those high in trans fats, refined sugars and processed foods. Consult with your doctor to see if your diet could be contributing to your chronic back pain and how you could change it. Maintaining a healthy weight could also help lessen your back pain by reducing the pressure on your spine.