Hello Tom,
My that is an unusual request/requirement. .004 is the min I like, it can go down to .003. Below that... I wouldn't trust it. Actually the "ring" configuration you speak of is unique enough that I believe it will be problematic. I would not do it.
As to the statement of board warping due to soldermask on one side; I disagree. Soldermask, especially LPI is pretty friendy stuff relative to warp. Unless there is some unusual condition of board thickness or finish, one side solder mask will not warp a board.
Brad Saunders
Coretec Boston Office
781 858 0783