I Googled and found a few trees for Charles Erasmus Mikkelsen at MyHeritage - one of them yours. I suppose that what you want is the Danish connection. Since there is a birth date and a location (Nakskov) known for Charles Erasmus it should be possible to find him in the Danish records, and go back to parents and grandparents etc. If it were Sweden I would make a try, but Denmark is another country :-)
I'm also thinking that he may have been a Carl Rasmus Mikkelsen from the start - adapted the spelling not just of his last name. And I'm quite sure there isn't a Mikkelsen family tree going back through the ages, since in Denmark, historically, people would have used patronymics, so that the generations would have shifted from Christian Rasmussen to Mikkel Christiansen to Rasmus Mikkelsen (as an example). I see that the father of Charles was also a Mikkelsen, so this family may have "frozen" the patronymic into a surname already - did the parents also emigrate?
It does not seem as if any Danish relatives have put a family tree on the Internet, but somebody familiar with Danish genealogy ought to be able to help.
I think this question should be tagged Denmark.