  Recent works have sought to realize certain families of orthogonal, symmetric
polynomials as partition functions of well-chosen classes of solvable lattice
models. Many of these use Boltzmann weights arising from the trigonometric
six-vertex model $R$-matrix (or generalizations or specializations of these
weights). In this paper, we seek new variants of bosonic models on lattices
designed for type B/C root systems, whose partition functions match the zonal
spherical function in type C. Under general assumptions, we find that this is
possible for all highest weights in rank $2$ and $3$, but not for higher rank.
