John Jay Research

I don't know much about non-Staunton sets. The chess museum calls the type you showed Cantonese "Burma" style. See here (look for Canton Bones):
http://www.chess-museum.com/bone--ivory--horn.html
Jon Crumiller has sets like that in his collection as well. You can find examples here:
http://www.crumiller.com/chess/chess_pages/chess_asian.htm
According to him, these were actually carved in China, not Burma (matches what Nic Lanier says at the Chess Museum). Here are 2 examples:
http://www.crumiller.com/chess/chess_pages/asian/Small_Burmese_ivory.htm
http://www.crumiller.com/chess/chess_pages/asian/BurmeseIvorySet.htm
Hoping you can help me in my current research (or point me to someone who can)!