Overriding a row FORMAT_STRING with a column FORMAT_STRING in MDX
After doing a bit more research, I found that the SOLVE_ORDER member property can get SSAS to use the format string from the MTD vs. Prev. MTD % member.

The following code worked for me:

MEMBER [ClosedDate].[Date].[MTD vs. Prev. MTD %] AS 
  [ClosedDate].[Date].[MTD vs. Prev. MTD] / [ClosedDate].[Date].[MTD],
  FORMAT_STRING='Percent', SOLVE_ORDER=1