I think the only way to do that would be to have a Folder for each part... not a great idea! ;-) If some generic information is shared amongst all files in the folder then it would be a good idea to map the Folder Card variable into the Part / Drawing data card.
Otherwise you would need to map the variables from the Part / Assembly to the Drawing. This only updates in SolidWorks so the Drawing needs to be opened, regenerated and saved AFTER the change to the Part / Assembly data card.
The variables on the card are also the properties in the model. The way we have ours set up is that after the model and its properties are done the drawing is created. When that happens it takes the properties from the model and adds them to the drawing. Once you save the drawing the properties fill out the card automatically. What you need to do is to make sure that the variable in the EPDM has a $PRPSHEET block with the slddrw extension. This would be in addition to the CustomProperty block. I use the same attribute name for both.
What you're asking for sounds very similar to what I've been asking for / looking for over the past 3 years. I ended up writing a very complex add-in to handle this. We have other document types (doc, xls, dwg, etc.) that all have the metadata linked. Please submit an enhancement request. Maybe we can get it moving.
It's possible to link a model and drawing using $prpsheet as the variable block name (see the help), but this doesn't cover all data entry scenarios.