Incorrect SUM number?

There will be a DAX formula that does it but I'm not exactly sure how (I'm new to DAX but not distinct count issues).

At a guess investigating SUMX might be a good start, it should be possible to get that to sum the distinct count for each month.

Here's a link to what looks like a similar technique:

http://www.powerpivotpro.com/2010/02/sumx-the-5-point-palm-exploding-fxn-technique/

Try create new key that combine SAPID and MonthID and then do DistinctCount on this new key field