I wish I could throw out some advise but I kind of have 2 conflicting views…and I am far from a doctor. I first think you should try to stay with exercises where your back is supported so you can reduce your chance of injury. Then I start thinking maybe you need to strengthen you back to help you out in the long run. I kinda think if you do decide to try exercises to try to strengthen your back then I would take it slow at first and see how your body responds, then you can try to build from there.
As for dropping 45 lbs…that is almost all diet. I would take a look at what you actually eat and look at areas where you can clean up your diet…eat healthier while reducing you calories. Keep in mind generally it is good to stay away from processed foods and try to keep it to wholesome healthy foods. When I diet down I like to cut some of my carbs down but not all…carbs are essential to staying healthy. I mostly try to cut out the white starches for the most part and replace them with lots of veggies. Try to eat 5-6 times a day but eat in moderation, so overall you will be consuming the same # of calories (but reduced to what you normally take in) but spread out through out the day…this will help with your metabolism. If you clean up your diet and take it seriously while trying to find ways you can stay active I think the pounds will drop off real quick. I would say a healthy rate to lose weight is roughly 2 lbs/week. That way you know you are not losing too much muscle mass. Hope this helps and good luck