From the recent discussion in the question you linked to, it seems that even (some? most?) finite-dimensional Banach spaces do not have categorical duals in the precise sense you describe. In fact it's clear what goes wrong here with the construction used in the algebraic setting of finite-dimensional vector spaces: the composition of the two maps you describe would be multiplication on $I$ by the dimension of $A$, but that is not a contraction unless $\dim A \le 1$!
You might have more luck with a category of Banach spaces and all continuous linear maps, although I do not know whether that also has a closed symmetric monoidal structure.