I've tried googling the word "portrait" in combination with various permutations of "cost," "price," or "commission" and "Edwardian" or "in 1912." Any variant of "cost" or "price" in those searches turns up recent auction prices and sites selling reproductions. "Commission" has just gotten very general mentions without any actual dollar or pound amounts attached. I've read the Wikipedia entries on John Singer Sargent and William Merritt Chase, and the material about Sargent on this site. I also looked at the Smithsonian's online collection of Sargent's letters, but found myself having an embarrassing degree of difficulty with his handwriting and was unable to determine if they contained anything relevant.
UPDATE: Thanks, folks! I've got an answer that's precise enough for my needs. I appreciate all the help and suggestions.