Yes, you can have the filler done while under anesthesia for the BBL. This is commonly done. However, I believe it is probably better to have filler injections when you are awake. It is important for the injector to see you face with at rest and with animation while he/she is injecting the filler.
There is absolutely no reason why you cannot have fillers done at the same time as your BBL. Make sure you discuss this with your doctor well in advance. Good luck from New York City