I figured out a workaround for this based on my needs. I was using linked PSDs, so copied and pasted the objects in place over the originals, went to Object > Clipping Path and created a clipping path based on the Alpha Channel, right-clicked on each object with clipping and converted the clipping path to a frame, repeated as necessary (3 objects for me). Next I copied all paths, merged all paths in Pathfinder, and hid the layer. Next I applied black fill to the unmerged paths, set the fill color to 1%, and applied the drop shadow. Then I grouped all paths and applied Knockout Group. Lastly, I cut the grouped paths, unhid the merged path, and pasted in place. This allowed the shadow to be inside the paths only and not the background.
I figured out a workaround for this based on my needs. I was using linked PSDs, so copied and pasted the objects in place over the originals, went to Object > Clipping Path and created a clipping path based on the Alpha Channel, right-clicked on each object with clipping and converted the clipping path to a frame, repeated as necessary (3 objects for me). Next I copied all paths, merged all paths in Pathfinder, and hid the layer. Next I applied black fill to the unmerged paths, set the fill color to 1%, and applied the drop shadow. Then I grouped all paths and applied Knockout Group. Lastly, I cut the grouped paths, unhid the merged path, and pasted in place. This allowed the shadow to be inside the paths only and not the background.