At this point I think the best bet would be to add the tag "Switzerland" to your question, to see whether someone who knows sources in Switzerland can point you to where one might find birth records from 1877 in St. Moritz.
In general, sources for both East Prussia and Courland are very spotty (there was a lot of destruction of records in both WWI and WWII), and what has survived has often not been indexed by the Mormons (and thus cannot be found via familysearch.org or ancestry.com). Thus, without knowing exactly where in East Prussia (or Courland) to look, it will be extremely difficult to find his family.