Let $X/\mathbb{C}$ be a smooth projective curve of genus $g>0$ (here $\mathbb{C}$ is any algebraically closed field, say of characteristic $0$). Let $S$ be a finite set of closed points of X and let $\operatorname{Pic}^{level,S}(X)$ be the (commutative group) scheme (of infinite type) parametrizing line bundles on $X$ with trivialization in a formal neighborhood of $S$. Is it true that every (global) function on $\operatorname{Pic}^{level,S}(X)$ is constant on connected components?

Here's an essentially equivalent problem, which is perhaps a little more concrete. Let $x\in X$ be a closed point and let $\operatorname{Jac}^{n\cdot x}(X)$ be the (commutative group) scheme of finite type parametrizing degree $0$ line bundles on $X$ with trivialization up to $n$th order at $x$ (a generalized Jacobian in the sense of Serre et al). For $n\geq 1$, $\operatorname{Jac}^{(n+1)\cdot x}(X)$ is a $\mathbb{G}_a$-bundle over $\operatorname{Jac}^{n\cdot x}(X)$. Is this bundle non-trivial?