Sandy- what church did the family belong to in Eau Claire? The closest in this area are either First Lutheran Church or Grace Lutheran. Lake Street Methodist is also a few blocks away. You might also try Lenmark-Gomsrud-Linn Funeral home for records. At the time you are looking, it would have been just Lenmark's or Lenmark and Son. Seems like quite a few Norwegian's used their services. The funeral home is still in service today.
For the death certificate, did you check under her maiden name? I know that isn't usual, but who knows. Was this a child stillbirth with midwife or physician? What was the hospital or was the infant's death at home. What was the cemetery used on the infant's certificate if that was filled in.
I think looking for a church record might be your best option. One more Norwegian Lutheran church would have been Our Savior's Lutheran. All the churches I mentioned are still in operation and have records going back this far and further.