I found a clue. The Archives of American Art at the Smithsonian are kind enough to lend microfilm reels via interlibrary loan, and on a reel I borrowed I found a manuscript of his that mentions using stearates. I had no idea how useful this could be to our conservator; if Hiler used zinc stearates, then they have a wealth of information at their disposal to address problems in the murals like this:
See the tiny, white splashy spots? They're not the result of someone spraying soda pop from a shaken can--they may be the result of zinc stearates in the varnish.
The conservators working on the murals were kind enough to take time out to show me exactly where the information I've provided them is helpful, and to explain many of the technical aspects of the information they need.
I still haven't found out which exact varnish Hiler used, but I did find out the composition of the paint: from the Ed Biberman Oral History at the wonderful Archives of American Art, he says:
But I had a friend, the very capable artist and technician Hilarie [sic] Hiler, who has written a book on art techniques. For the WPA Project he did a very beautiful mural in the Marina-is it the Marina Park in San Francisco? I believe it's just called "The Marina." I'd seen it there and was intrigued with its beautiful eggshell surface quality, the "sheen" that it had, and asked him what he had used. He told me that he had made a preparation of oil of spike, which is a very wonderful aromatic vehicle, with wax and oil colors. And being a very gracious man he gave me the formula for this little stew and I used it for the three murals that I painted. It is a very fine solution to one of the problems of having to paint a mural in oil.
--Oral history interview with Edward Biberman, 1964 Apr. 15, Archives of American Art, Smithsonian Institution.
There's a photo of the dramatic difference between the cleaned and uncleaned portions of a section of mural that I put up on Flickr, during that wonderful visit with the conservators at the Museum.
And if anyone knows what varnish Hiler used, do let me know!