Start here: confirm what your own health system provides
Where fat grafting is reconstructive, correcting a defect from cancer surgery, trauma, radiotherapy or a congenital condition, it is usually funded by health systems that fund the underlying reconstruction, since it is treatment of an acquired deformity rather than cosmetic enhancement. This is particularly true following breast reconstruction, where fat grafting to refine contour or correct asymmetry is a well-established, frequently funded part of the reconstructive pathway.
Where fat grafting is being used to enhance normal anatomy rather than correct a defect, for example general facial or breast augmentation, it is treated as cosmetic and is not funded. Confirming which category applies to your situation is the first step before comparing any private price.
Congenital conditions causing asymmetry, such as Poland syndrome or facial asymmetry from hemifacial microsomia or Parry-Romberg syndrome, are also generally treated as reconstructive, and fat grafting used to correct them typically qualifies for the same funded pathway as any other reconstructive correction. It is worth naming the specific underlying diagnosis clearly when asking your health system what is covered, since a vague description of "asymmetry" can sometimes be assessed differently from a named congenital or acquired diagnosis.