My doctor told me CoolSculpting was ok to do while I'm breastfeeding.

You should wait till your done breastfeeding . After pregnancy your uterus has enlarged and will take time to reduce in size, you have hormonal changes that haven't stabilized and your abdominal musculature will change.
If you are concerned about the treatment having determental effects on lactation for your baby, no one knows the answer because the treatment has not been tested on pregnant or lactating women and probably never will. Retrospective studies may eventually be done but at this time there are no recommendations except to avoid treatment and to err on the side of safety.  Basic intuition would suggest cold exposure should not have determiental effects but there is no way to accurately answer this issue at the present time.