One thing I did notice. I found this in the New Hampshire Birth and Christening Index on Ancestry. Hermann's ethnicity was listed as Canadian, which his mother and August were not, suggesting that who ever Herman's dad was, was from possibly out of country? I think you are going to find it difficult to find Herman's bio dad. Who ever he was, even Katie didn't even know where Herman's bio dad was born, most likely she didn't know him very well (eg. local family)

On one of censuses you posted you will see the notation "M1" next to Katie's name. That indicates that she stated she had been only married once, to August. So I don't think she had been married before.

New Hampshire, Births and Christenings Index, 1714-1904
about Herman Vattes Name: Herman Vattes
Birth Date: 26 Feb 1898
Birth Place: Manchester, Hillsborough, New Hampshire, United States
Gender: Male
Race: White
Ethnicity: Canadian
Father's Name: August Vattes
Mother's Name: Kathe Liebing
FHL Film Number: 1001050