The notion that fat grafting would fix hypertophic scarring is anecdotal at best. I have tried it 6-8 times and saw no difference.
In my opinion, fat grafting would not improve hypertrophic scarring. Your present therapy of v beam and steroids is an excellent one. If you would like to consider remodeling of the skin, consider PRP injections. In addition 5-fluorouracil and verapamil are also helpful.
Fat grafting may help improve the quality of skin in some conditions. Since your scar is pretty new, it is best to allow it to heal over the next 6-12 months. Check with your doctor for scar care at this point.
I would not get too aggressive at 6 weeks post-procedure. You may do just fine with the treatments that you are presently having done. Nonetheless, fat grafting can be very helpful insofar as the fat does bring mesenchymal stem cells into the area. Those stem cells can have a very beneficial effect on the overlying skin.