The safety is many cosmetic procedures during pregnancy and breastfeeding is not well established. Why? Because they are cosmetic treatments, meaning not medically necessary, and we as a medical community tend to not test non-medically necessary treatments on women of such status.
That said, the placement of lip filler is a fairly straightforward procedure. Generally, no sedation is required and risks are minimal to the patient.
As long as patients are otherwise well with no illness, symptoms, or signs of mastitis/acute breast inflammation, dermal filler placement is argued by many physicians to be safe while breastfeeding.

