Category of $\mathfrak{sp}(2n)$-modules with bounded weight multiplicities

Let $\mathfrak g$ be a finite dimensional simple Lie algebra. Denote by $\mathcal B$ the category of all bounded weight $\mathfrak g$-modules, i.e. those which are direct sum of their weight spaces and have uniformly bounded weight multiplicities. A result of Fernando shows that infinite-dimensional bounded weight modules exist only for $\mathfrak g=\mathfrak{sl}(n)$ and $\mathfrak g=\mathfrak{sp}(2n)$. If $\mathfrak g =\mathfrak{sp}(2n)$ we show that $\mathcal B$ has enough projectives if and only if $n>1$. In addition, the indecomposable projective modules can be parameterized and described explicitly. All indecomposable objects are described in terms of indecomposable representations of a certain quiver with relations. This quiver is wild for $n>2$. For $n=2$ we describe all indecomposables by relating the blocks of $\mathcal B$ to the representations of the affine quiver $A_3^{(1)}$.