Asked around. We do perinatal for no charge for anything except a vault to protect the coffin and a headstone. We waive the fees for funeral directors, embalming, transfer of deceased (to FH, to church, to cemetery), visitation, graveside service, casket, family limo and provide a free plot in our cemetery (with opening and closing fees for the plot also waived).
We're uniquely generous with this stuff in the area (we're very high end) but I would be surprised if there weren't comparable companies in other areas. I imagine it's viewed as excellent PR and a good investment in the rest of the family at a business level. For perinatal you ought to be able to get the works without spending a cent, although the nominally "cheaper" funeral homes may still charge. Clergy and choir often have an honorarium for their services but it's voluntary and if your family are good, tithing members of the congregation then I can't imagine any expectation that they would pay at a time like this. Total costs should be close to $0 if you know what you're looking for, as with anything else a lack of familiarity with the system will cause you to spend inefficiently.
Absolutely make arrangements for the funeral or work with the grandparents to do so. Medical bills can wait but the funeral can't and the family have each other, and their older child, to look after at this incredibly difficult time. If you think you'll do a more competent job than the grandparents in making arrangements and getting a good price with a funeral home/cemetery then make the arrangements and have them write the check. Worth being aware though that you may be committing the family to a particular cemetery if you don't move around a lot, it's likely the parents would like to be near their baby when their time comes so pick somewhere nice.
Then offer what you can afford towards medical having done the lion's share of the funeral expenses with knowledge and effort alone.