Don't personally know what a fat burner injection is, but if docs say no what's your question? Anything you consume your child consumes.
I think this procedure has, unfortunately, been under scrutiny for some time. They don't dispute it can work but at what costs? Things that can happen are infection, disfigurement, fatty liver, kidney problems, tissue death, etc. Not good stuff and the procedure is banned in some countries due to safety. https://www.webmd.com/beauty/features/fat-busting-injections-under-scrutiny#1 Have you had it done previously before the baby? I also think you need to be your best health self and since there are some known issues that can arise, I wouldn't do it while nursing. I personally wouldn't do it at all. :>) I know it is less than some cosmetic surgeries but I don't think it is worth the risks.