Sharp, shooting pain on upper back/side of left thigh after BBL?

It sounds like you may have sciatica after your BBL.
This can happen from time to time.
Sciatica develops when there is pressure on either nerve roots or the sciatic nerve.
Pressure on the sciatic nerve often comes from the nerve roots as they exit the spinal cord but can also be from pressure in the buttocks or any where along its course.
If fat is grafted very close to the nerve than pressure from sitting can cause excessive pressure on the sciatic nerve.
Most plastic surgeons are very well educated and surgical anatomy and should avoid grafting large amounts of fat in the proximity of the sciatic nerve.
Even so adding volume anywhere in the lower buttocks can cause excessive pressure on this nerve.
Follow up with your plastic surgeon and have him or her address your concerns.
See the included link for a brief explanation of sciatica
Best, 
Mats Hagstrom M.D.

Pain after BBL can be a common phenomenon as the process of fat grafting into the buttocks requires use of grafting cannulas that can irritated sensory nerves; Your nerve pain will continue to improve over time; sometimes it may take upto six months.