you are right when you think there are two lines of Andersen in Klægsbøl (nowadays Klixbüll in the north of Germany).
Anders Sønnicksen was ennobled march 29 1452 and his descandents take the last name "von Andersen" after that time.
Momme Andersen born 1484 in Tønder was a merchant and town councillor, thats the meaning of raadmand.
One of his sons was:
1. Anders Momsen merchant and town councillor in Tønder. His wife was may be named Margaret. After killing a nobleman in self-defence on sunday the 8 july 1554 he had to leave the country. He died 1559 in foreign abroad.
The only child of him who is named is: Thomas Andersen merchant in Tønder. Died may 2. 1588, 45 years old. He was married with Catharina Jensen died september 8. 1608, 56 years old.
they had 5 children
1.Marina, died 11/12/1643, aged 70 years, married 1589 with Lorenz Petersen died 6/23/1618. 9 children
2. Andreas Thomsen the younger, born 1576, died 6/22/1624 married with Magdalena de Behr, died 8/23/1661 aged 85 years. 6 children.
3. Catharina, born 8/12/1581 died 6/8/1624, married 1597 with Lübbert Wydow, died 5/26/1628. 8 children.
4. Anna, died 3/14/1615, married with Simon Petersen, died 12/7/1615. 5 children.
5. Hans Thomsen, born 1587, died 4/26/1650, married with Christina (Tüchsen), born 7/15/1590, died 3/16/1667. May be 8 children.
I hope this Information will help you a little bit to clear. Ancestors of Momme Andersen are un known. Churchbooks will not be available for that time.
It is very unlikely that there is a relationship between Momme Andersen and Anders Sønnicksen.