At the moment it's a component clearance rule, but I'd like to know as a general question if there's a way to do this. The rule in this particular case works exactly as intended with the ClassName applied in the component properties, no problems on the rule side of things.
For the record, I also tried using a PCB Rule directive on the blanket covering the components in question, but I just wind up with a bunch of rules that target all of the nets contained in the blanket, and nothing targeting the components.