I originally thought that on one had discovered this peculiar story before, but Google books helped me to discover Axel Paulin's Svenska öden i Sydamerika (1951). Paulin's list of sources is very impressive, but he is often short on specific references. The story of the Chilean princes is described like this:
When going through some documents from the Swedish Field Chancery during the Polish period a letter from the Swedish envoy in the Hague was following letter was found. Paulin then goes on to quote from Palmquist's letter dated 4 March 1705 as well as from a the translated summary of Colonel Scott's letter attached to this report. Paulin apparently never looked among Palmquist's drafts and so he never found Scott's original proposal.
Paulin's conclusion was that there must have been some real basis for the proposal. He suggests that some of the cargo was coming from the East Indies, but finds no reasonable explanation for the 27 princes.
Paulin tried to follow the story and noted that Charles XII ordered Palmquist to keep the authorities in Stockholm updated. The King also instructed the Chancery College and the College of Commerce to discuss the matter and send their views to him.
Oddly enough Paulin concludes by saying that no further traces of the story could be found beyond the fact that the King's letters did reach Stockholm. My distinct memory is that there was in fact plenty of evidence, especially in the archive of the Chancery College. If I can find my old notes on the matter I'll add a part 4.