Put simply, "the divorce paper's being his only leverage" is incorrect. There is no need for the possessive.
With the divorce paper being his only leverage = the divorce paper is/was his only leverage.
Whether "being" means "is" or "was" (or even "will be") depends on the tense of the relevant part of the piece.
Let me ask you a follow up question because I am still confused about the grammatical rule which makes "divorce paper being his only leverage" possible.
Is the grammatical usage of "being" in the above phrase the same as the usages of "being"s in the following sentences?
"They objected to the youngest girl being given the command position."
"The mistakes I made had nothing to do with it being the first game."
If not, can you shed some light on how their usages are different?
Last edited by rainous; 16-Dec-2011 at 03:14.