You may need a macro to accomplish what you are after. I don't do that exact same thing but I use a macro to pull info from the drawing file name and assign it to the drawing custom properties. I don't know for sure but I think a macro could probably also pull the info from the "Model in view specified in sheet properties" and then assign it to drawing custom properties. You would need to manually run the macro, so it isn't as automatic as you may want, but you could actually make the macro do all that and save, and then assign the shortcut ctrl+s to run the macro, so it will update the drawing custom properties based off of the model in it every time you save.
YOU CAN ACHIEVE IT THROUGH "PROPERTY TAB BUILDER"
START > ALL PROGRAMS > SOLIDWORKS 2015 > SOLIDWORKS TOOLS > PROPERTY TAB BUILDER
YOU'VE TO CREATE FOR BOTH PART AS WELL AS DRAWING. IF THE SAME PROPERTIES FOR BOTH PART AND DRAWING. FIRST CREATE PART PROPERTTIES AND SAVE THE FILE(.prtprp). ONCE AGAIN OPEN THE SAME FILE IN PROPERTY TAB BUILDER AND CHOOSE THE TYPE AS DRAWING AND SAVE THE FILE(.drwprp)
THEN PUT THOSE TWO FILES LOCATION IN
OPEN "SOLIDWORKS" > "OPTION" > "FILE LOCATION" > "CUSTOM PROPERTY FILES" > CLICK "ADD"> SET THE FILE LOCATION OF CUSTOM PROPERTY FILES.
Not at work right now, so no SW in front of me to test my theory, but it's simple enough for you to try, if you wish.
In your drawing's properties dialog box, for "Description", try entering $PRPSHEET:"Description" in the Value column.
I'm thinking this will force the drawing to go get the referenced part/assembly's Description value and assigned to the drawing's Description.
Plus, it's dynamically updated and it's not hard-coded to any particular part/assembly name.
I believe this method works for the "SW-" type properties, so I'm hoping it will work for your situation as well.
If my suggestion fails, try putting an equal sign before the "$" to turn it into an equation.
This works and evaluates great, but ONLY while the drawing is opened, in EPDM the DataCard will still read even if the drawing preview shows the evaluated string.