A final comment and then I'll wait to hear from you with more details of this family genealogy.
You researched the Thingvalla Line, so you know it was a direct line and took immigrants transatlantic from Copenhagen to New York. No stopovers. Whoever P.C. Nielsen was, this was no doubt the route he/she took.
Many Danish immigrants left from Copenhagen but took a feeder ship to Hull, England, then a train to Liverpool, and from there a transatlantic ship. Other Danish immigrants embarked at Hamburg. Ports of entry to the USA included Philadelphia and Boston. Knowing you can narrow your search to the Copenhagen-New York immigrant route might help a little.