the carbs for one shake is roughly 14grams (at a guess) and you can only have 20grams a day on atkins induction. of which most of them should come from veg... so you are going to have very little veg!
i would say no because although both are ketogenic diets they have very little in common besides this.
atkins = high fat low carb
cd = low cal low fat low carb
they wont work together and allow you to benefit 100% from either plan
i would start on cd and then look to do an atkins form of low carbing to maintain the weightloss.
hope this helps... x