I don't have the definite answer to your question, but I found a mortuary notice for a Barnet Karper in the Philadelphia Inquirer of Friday, January 14, 1916.... (you note he died in 1910)
KARPER -- Jan 11, 1916, BARNET II, husband of Frances A. Karper, in his 65th year. Relatives and friends of family are invited to attend the funeral services, Friday, at 7:45 P.M. at residence of his daughter, Mrs. R. Rappaport, 5634 Hunters ave. Interment Wiconisco, Pa., Saturday.
Here is a maybe for Francis and parents in Lancaster PA, in 1860 - indexed as Henry SHERBAHN, but hard to determine if that is the actual spelling on the image. Household includes (assumed) wife Ann, as well as some Sherbahn children including Frances, age 9.
1870 census for this family shows Ann as head of household, and is indexed as SHERBALIN, but looks more to me like SHERBAHN, Frances is included at age 20.
If this is the right family, a probable obit for Ann was found in the Patriot newspaper of Aug 10, 1877:
Mrs. Anna B. Sherbahn (spelling of last 2 letters a little blurred), widow of the late Henry Sherbahn, formerly proprietor of the Old Cross Keys hotel, Mount Joy, died at the residence of her son-in-law at Middletown recently. The remains were buried at the Bethel church, Mount Joy.