Syringe vs pump BBL: Do doctors still use the syringe method for fat transfer?

The issue will not be the pump or the syringe but the size of the injection cannula and the depth of injection.

Great question! There is a lot of conflicting information online, indeed, and good Brazilian Butt lift results really are highly dependent on the surgical techniques employed during harvesting, processing, and reinjecting of the patient's fat.
While this is a somewhat over-simplified short explanation, here are the basic surgical techniques I have employed for the last 15+ years with fantastic results: 
For harvesting, I use suction assisted tumescent liposuction using slender, 3mm cannulas with 1mm holes. The small cannula holes allow me to collect only the smallest fat particles, which make the best fat grafts. This step also contours the waist and hips, adding to the final results.
The harvested fat is then carefully processed and purified by decanting and centrifugation. Only a fraction of the donor fat removed will actually qualify for re-injection.
The purified fat is transferred to the buttocks by injecting small amounts (micro fat injections) into different areas and at various depths in the subcutaneous region, again, using very specific cannulas. Correct injection methods are essential to achieving smooth, uniform, natural looking results.
Fat grafting is a highly technique-dependent procedure, and some surgeons are having more success than others. You can successfully research and compare! (See the web reference for 6 questions you can ask plastic surgeons during BBL consultations.)
Best wishes!

Hello, thanks for your question and provided information. every surgeon has it's own manners, consult with your board certified plastic surgeon. Good luck!
Advantages:
One treatment may be all you need to reshape problem areas
Treat more areas—and larger areas—in one procedure
May be performed with either local or general anesthesia
More consistent and predictable result than noninvasive treatments
Liposuction, sometimes referred to as "lipo" by patients, slims and reshapes specific areas of the body by removing excess fat deposits and improving your body contours and proportion.
Different areas of treatment:
Thighs
Hips and buttocks
Abdomen and waist
Upper arms
Back
Inner knee
Chest area
Cheeks, chin and neck
Calves and ankles
Liposuction can be performed alone or along with other plastic surgery procedures, such as a facelift, breast reduction or a tummy tuck.
What liposuction can do?
Liposuction is not a treatment for obesity or a substitute for proper diet and exercise.
It is also not an effective treatment for cellulite—the dimpled skin that typically appears on the thighs, hips and buttocks—or loose saggy skin.

The technique is very important to ensure maximal survival of your fat graft.  I use an excellent system from Wells Johnson to optimize the procedure.