I just skimmed your definitions so I might be getting something wrong. But the idea is that your symbolic reasoning system, prior to interpretation, corresponds to a system of propositional schemas, in which the logical form and logical dependencies of the propositions have been specified, but the semantics has not. Meanwhile, the facts about the world are described by another, bigger system of propositions (whose semantics is fixed). The search for groundings is the search for subsystems in this description of the world which have the logical structure (form and dependency) of those in the uninterpreted schema-system. The stuff about hypergraphs was just a guess as to the appropriate mathematical framework. But maybe posets or lattices are appropriate (for the deductive structure).

I haven't thought about intension vs extension at all. That might require a whole other dimension of structure.