Let $\mathcal{C}$ be a monoidal category. There is a notion of Morita equivalence of algebra objects internal to $\mathcal{C}$. Does each Morita class have a symmetric Frobenius representative? A Hopf representative?

Following arXiv:math/0111139, suppose $\mathcal{C}$ is semisimple and rigid with finitely many irreducible objects and irreducible unit. Each indecomposable semisimple module category over $\mathcal{C}$ is equivalent to one of the form $\text{Mod}_\mathcal{C}A$ for some algebra object $A$ internal to $\mathcal{C}$; such algebras are called indecomposable semisimple, and their Morita classes are in bijective correspondence with such module categories. In this language, my question is whether, given such a module category $\mathcal{M}$ over $\mathcal{C}$, there is a symmetric Frobenius (or Hopf) algebra object $A$ in $\mathcal{C}$ such that $\text{Mod}_\mathcal{C}A$ is equivalent to $\mathcal{M}$.

If not, are there conditions on $\mathcal{C}$ under which $A$ exists?