About this: "To English eyes it looks like an 'a' so he probably got tired of telling people how the original spelling was pronounced and just called himself Martin." Is there evidence that Charles's father, Martin/Morten, left Bornholm? I would presume he did not emigrate (until or unless we find that he did) and so would not have had any need to Anglicize his name.
Would have been nice to have found Charles/Carl on the church book list of person's leaving Rønne.
Pauline - I notice you added some of the new info to your Ancestry.com family tree. Nice! For Carl's father's birth place you have "Græshave, Denmark." You might want to correct that to the full, correct place name (often times helps to avoid confusions), which is Græshave, Lollands Sønder, Maribo, Denmark. You note his mother as "Karen Mortensdatta." That would be Mortensdatter. This means she was the daughter of Morten. There is no patronymic name ending "datta."
Stay in touch and let us know if you find more of Charles's movements. Hope so!