There are a few other considerations here that are worth taking into account.
Jews in the Pale didn't traditionally use surnames, instead referring to themselves with the equivalent of "Joseph, son of David". They were forced to take up surnames in the first decades of the 19th century, names that were often linked to occupation or characteristics.
But Jewish lineage goes back much farther, especially in the case of the kohanes and Levys who trace themselves to Aaron. Families identified much more strongly with these tribal associations than with their new surnames. Thus when moving to America, people often chose names that reflected their tribe, not their (recently acquired) surnames. Hence names like Cohen, Cohn, Levine, etc.
This is the case of my own family.
Records show Yosiel Garbarski marrying Tyla Kac in the Lithuanian town of Balbieriskis (Balbirishuk in Yiddish) in 1860. They then moved to Kaunas/Kovno sometime in the 1870s.
(Garbarski means son of a tanner - and indeed we find his dad recorded as a tanner in 19th century, though Yosiel never took up the profession)
They moved to Iowa 30 years later as Joseph and Tillie Cohen.
We know from family interviews, tombstones, and US temple records that Joseph was proud to be kohane, and that his kids spoke often of being "Cohens on both sides".
One last thing - I haven't found many/any records of Litvaks using 'Cohen' as a last name in the 19th century. Lots of Katz's, but not Cohens. With that in mind you may consider that your "Max Katz" is searchable by his surname (also spelled Kac, Kats etc). But that Anne Cohen may have had a different name.
Best of luck - sounds like our Cohen + Katz stories are pretty similar!