I made a new part to try this out on, to test if I understood it right. Correct me if there's an effective better way or a workaround other than recreating it entirely.
We should be able to suppress a dimension in a sketch in one or more configurations. Currently, you can set a dimension value to specific configurations. If you change over to the un-selected configurations, the dimension still appears, just not at the specified value. Setting a dimension to driven does not act as suppressing it in the desired unused configuration only, but rather setting it to driven makes it driven in all configurations. Therefore, suppressing a dimension ought to set it to driven in either This or Specified configurations.
Thanks to Rob Edwards, whose idea Ability to Rename Sketch Relations reminded me of this issue, because it exhibits suppressed / unsuppressed control of sketch relations. Also, this thread here was a good idea too, easily enacted.
Here's an example, simplified as I could. One design I have for a chemical tank is being produced by two different fabricators due to materials of construction outside of our preferred fabricator's capabilities. Each of the fabricators have returned engineered drawings from our design, with their own form and layout of lift lugs on the domed top of the tank. One design defines its location radius dimension upon where the lug's center point meets the dome, and the other defines its location radius dimension to the center of the lift lug hole. Because I had one sketch to define layout locations, and other sketches converted or related those entities to create weldment features, I then had to fork my layout sketch into a separate one and recreate all of the subsequent related features from a different source. If I had been able to suppress one dimension in a material/volume config, I could apply one diving dimension in first condition, and a differently defined driving dimension in the second condition.
Tom has marked Rob's answer as Correct