We study the computation of our recently introduced Whitney polynomial and the enumeration of the spanning hypertrees for hypermaps whose hyperedges have length at most $3$. This is a class of hypermaps where the computation of the above invariants depends only on the underlying (multi)hypergraph structure. We develop deletion-contraction formulas involving six types of generalized loops and bridges, and we prove results on special substitutions into our Whitney polynomial. We generalize the reliability polynomial and the random cluster model to hypermaps in general in such a way that they can be computed using our Whitney polynomial. Finally we explicitly count the spanning hypertrees in reciprocals of plane graphs in which every vertex has degree at most $3$.