I was thinking for instance about smooth functions on a compact group, or more generally smooth functions on a compact monoid in the category of smooth manifolds, which form, if I'm not mistaken, a bialgebra in the category of Frechet spaces with the projective tensor product.

I would like to add a comment about your sentence "this leads me to believe that the operads themselves should be equivalent." Be careful, the fact that two categories of algebras are equivalent does not imply in general that the operads themselves are weakly equivalent. The situation is analogous to what happens in representation theory: Morita equivalence does not always imply equivalence.