I have heard of doctor's offices doing that, but in my area I have never heard of any success in having a doctor's office reduce the rate for uninsured patients. A doctor's office who is willing to do that actually could be prosecuted for insurance fraud, federally (if they participate in federal programs) and in most states (WA has specific laws on this for example).
Around here, to cover a family, it costs an average of $1,200 per month in insurance premiums. In WA CS laws it cannot total more than 25% of the CS to add the child otherwise you are not required to.
For BM in our case that amount would have been about $80 a month. She did have available to her in KY individual plans that would have cost her $40 a month to cover SD. She was out of compliance with the court order.