I do not know much about the Tannaka-Krein duality itself. But regarding the last part of your question

Also if somebody could cast some light on possible generalizations of this proposition (to the case of Hopf algebras in braided monoidal categories)

i think that bosonization/transmutation and reconstruction methods between hopf algebras and hopf algebras in braided monoidal categories, may be closely related to what you are looking for. Roughly speaking:

The **bosonization** method consists of the following idea: Given a hopf algebra $A$ in the braided monoidal category $\mathcal{C}$ one can construct an ordinary hopf algebra $\mathcal{A}$ such that the category of the $A$-modules in $\mathcal{C}$ (i.e. the braided modules of $A$) and the category of (ordinary) modules of $\mathcal{A}$ are in a bijective correspondence which provides an equivalence of categories (between the corresponding module categories).

In the case that $\mathcal{C}={}_{H}\mathcal{M}$, where $H$ is some quasitriangular hopf algebra i.e. in the case in which the initial braided monoidal category is the category of representations of some quasitriangular hopf algebra $H$, then it can be shown that the (ordinary) hopf algebra $A=A\ltimes H$ is a smash product hopf algebra.

A special case of the above has proved particularly interesting for physics: If $H=\mathbb{CZ}_2$ then the hopf algebra $A$ in ${}_{\mathbb{CZ}_2}\mathcal{M}$ is a hopf superalgebra and the smash product hopf algebra $A\ltimes\mathbb{CZ}_2$ is an ordinary hopf algebra with categorically equivalent representation theory to the initial superalgebra.

At the level of physics this can actually be understood in the sense that the hopf superalgebra and the smash product hopf algebra produce the same physics. In other words, SUSY hopf algebras can be substituted by suitably chosen ordinary hopf algebras retaining the same physical interpretation.

The converse of the bosonization technique is the so-called **transmutation** method. You can find these methods (in their full generality) in: Majid S., Cross Products by Braided Groups and Bosonization, J. of Alg., v. 163, 1, 1994, p.165-190.

On the other hand there are also various **reconstruction** theories and methods. In the frame of hopf algebras in braided monoidal categories, this amounts to the following idea: Given a braided monoidal category $\mathcal{C}$, we can reconstruct a quastriangular hopf algebra $H$, such that the representation category ${}_{H}\mathcal{M}$ of $H$ is isomorphic to the initial braided monoidal category $\mathcal{C}$.

A nice bibliographical/historical review of this and related ideas can be found in ch. 10, sect. 10.4, p. 201-203 of S. Montgomery's book (*Hopf algebras and their action on rings*). An example of a (very) simplified reconstruction theorem is provided by theorem 10.4.2, p. 199 of this book.

Finally, i think it would be interesting to have a look at ch. 9.4, p. 467-499, from Majid's book "Foundations of Quantum group theory".

Tannakian categories(theorem 2.11) $\endgroup$ – Denis Nardin Sep 16 '19 at 21:15