We study Schubert calculus in the torus-equivariant quantum $K$-ring of the Lagrangian Grassmannian $\mathrm{LG}(n)$. Our main tool is the $K$-theoretic Peterson map due to Kato. The map is from the (localized) equivariant $K$-homology ring $K_{*}^{T}(\mathrm{Gr}_{G})$ of the affine Grassmannian $\mathrm{Gr}_{G}$ of the symplectic group $G=\mathrm{Sp}_{2n}(\mathbb{C})$ to the (localized) torus-equivariant quantum $K$-ring $QK_{T}(\mathrm{LG}(n))$. We determine explicitly the kernel of this map.