Without appearing too facetious, there likely was not a German family that did not use the male names of Wilhelm or Carl during the 1850s.Very likely, Carl was the baptismal name of the son you give.It would not surprise me if the name Carl would be found seldom used as his given name after he arrived in America.However, if the name Carl was used, very possible would be that Carl and Charles became interchangeably used.I have seen this with other German immigrant families. My suggestion is to study the webpage: http://www.kerchner.com/germname.htmhttp://www.kerchner.com/germname.htm entitled “18th Century PA German Naming Conventions”.
Do you know any descendents of the wife and husband (Anna and Christoph)?If so, I suggest that you work backwards from those known descendents.Obituaries and US Census data are very good sources to understand the given names used by individuals.
If working backwards cannot be done, then you really have one recourse, it seems, other than census indices.That would be to possibly find this family on a passenger ship list.However, that will not likely indicate where they eventually settled if they can be found this way because so many gave their reason for travel as ‘Staying in America’.
During the 1850s two primary locations to settle at first would have been Missouri or Wisconsin.Frankly, I am not aware of a family with these names settling in Wisconsin for any length of time in the 1850s so I recommend you try Missouri, if you feel right about that location.Since you know they are from Mecklenberg look for areas settled by immigrants from Mecklenberg.Foremost, do not leave anyone having the surname Heidle/Heidel overlooked who can be found to have come from Mecklenberg until it is fully understood what all their ‘first’ names were.
As far as Heidels that I have become aware, several from Mecklenberg settled in Kansas and became Civil War cavalry veterans.However, I am not aware that a wife of the one named William was Anna, but Anna could likely have been her baptismal name and typically not used in legal documentation, as pension or census records would be considered.