From what I understand, the surname HAUSCHILD did originate in Hanover.
You may want to look at Federal Census Records1900, 1910, & 1920 for Henry HAUSCHILD I or Henry HAUSCHILD II.These census records ask when; if foreign born- how many years they have lived in this country & where they came from.Then you can look within approx. 5 yrs. (or more)after they came to the country for naturalization papers which could provide more information.
Before you look at the census records , IF you do not have the family's home address you may want to look through city directories, soyou can find where the family lived & determine the enumeration district.(this is how the census records for the early 1900's were organized).
Have you ever tried to see if you can access Henry II's employment records?This could be helpful.