I’ve spent another hour on this, and the only way I can get the pad addition method to work is if the through hole is plated. I did have some concerns about tolerances after through hole plating, and agree that the holes should be non-plated ideally. Unfortunately, it seems that I can’t assign a pad number to an NPTH, so whenever an NPTH intersects the SMD pad the DRC throws an error. I’ll switch back to my “good enough” design with clearances.
If an option existed for subtracting graphic items from each other, this would be an incredibly easy task 