I agree with the core exercises mentioned above. Here are a few more ideas that I thought of for you:
1. Your client will gain core strength from various free weight exercises done standing up. Getting a client to do more free weight exercises rather than machines will help them strengthen their core muscles. Often times we don’t realize that our stabilizing abdominal muscles are hard at work while we are doing other exercises such as squats, lunges, and even arm work sitting on a stability ball.
2. Some good core exercises like wood chops or torso rotations can be done standing up with a medicine ball, cable pulley system or resistance bands.
3. Planks may still possible. One idea would be to have the client plank at an incline. This takes some of the body weight off them yet strengthens their core. I had one client who was 257 pounds, and she was fine at an incline (for example she put her elbows on a bench instead of the floor for plank). Each client is different, though.
4. Simple kickboxing moves are great for the core! Throw in some simple drills for cardio like jabs and punches or maybe some knee lifts and this will help with core strength. I could give you some specific drills if you like–this is a format that I teach and it is amazing how much it works the core.
I hope some of these exercises help you and good luck!