Was there a hospital room for Elizabeth? I know for the Mary reveal Cass and Adam found evidence that Mary McKinnon had been in a secret hospital room that Reginald had bizarrely kept untouched since Mary's accident. I don't remember the Elizabeth murder.
The cellar room was a bedroom where Donna had to hide for months that conceal had pregnancy. She did give birth there but it wasn't a hospital room (at least not in 1984 when she hid Catlin in it).
I'm not sure that I would rely on a black market baby doctor to commit a murder for me. They seem like different specialties.