Please disregard Arjun's responses as I believe they are AI generated.
I found a solution to the issue on my own. First off: I believe that what I ask in my question can not be achieved using referenced subsystems. But it is possible to do it using referenced models instead. I found this article which paved the way for me: Configure Instance-Specific Values for Block Parameters in a Referenced Model
So what I did was convert my referenced subsystems to referenced models, and then defined Simulink parameters in the Model Workspace to be used for choosing the variant. You need to tick off the 'Argument' box, which allows the parameters to be defined instance specifically. Remember to type default parameters, Then, the parameters can be modified in each instance of the referenced parent subsystem through the Model Data Editor. Note that this approach needs the 'variant activation time' to be on startup.