Update:
Robert and Martha Bourzutschky of Grand Rapids, MI formally changed their name, and that of their children, Frank, Charles and Mary, to GILMER on April 21, 1881 by Local Act No. 358 (Public and Local Acts of the Legislature of the State of Michigan, p. 229).
The family last appears as Bourzutschky in the U.S. Census of 1870 in Milwaukee, WI, Ward 2 and in the U.S. Census of 1880 in Grand Rapids, MI.