A lot of people have the idea that the Copperheads were the conservatives of their day. That wasn't always true. A lot of old radicals and laborites supported the Democrats, and more or less gave some aid and comfort to the Confederacy. David Goodman Croly, a radical married to an early feminist, accused Lincoln of promoting "race mixing." His son, Herbert, went on to be one of the founders of the liberal New Republic magazine.
Lincoln carried Wisconsin 56%-44% in 1864, and got the same 56% in 1860. His numbers were pulled down I suspect by the substantial numbers of Germans in Wisconsin, particularly German Catholics, the latter of which simply didn't vote GOP at all.