To my knowledge there are two main approaches to categorify the notion of a vector space. I will refer to them as BC-2-vector spaces (Baez, Crans) and KV-2-vector spaces (Kapranov, Voevodsky). Both define a symmetric monoidal bicategory (denoted by $\mathsf{2Vect}_{BC}$ and $\mathsf{2Vect}_{KV}$, resp.) and in both we can take direct sums, but a significant difference between them is that we don't have an additive inverse functor $-:V \to V$ for a KV-2-vector space, while for BC-2-vector spaces we do.
In particular, this means that for a KV-2-vs $V$, we don't have an antipode on the tensor algebra (or whatever the correct 2-categorical name is) $TV=\boxplus_{n \geq 0} V^{\boxtimes n}$. For me this is a problem, since I would like to categorify the notion of a Nichols algebra.
On the other hand, for an abelian group $G$ in the KV-picture we can easily categorify the category $\mathsf{Vect}_G$ of $G$-graded vector spaces to a bicategory $\mathsf{2Vect}_G=\oplus_{g \in G} \mathsf{2Vect}_{KV}$ and just as MacLane's third abelian cohomology classifies braided/symmetric monoidal structures on $\mathsf{Vect}_G$, the fourth abelian cohomology classifies braided/symmetric/sylleptic monoidal structures on $\mathsf{2Vect}_G$. This is now again very helpful, and actually the reason why I prefer the KV-picture.
So my question is: Is there another way to define a tensor algebra of a 2-vs $V$ in $\mathsf{2Vect}_{KV}$, so that it will be a Hopf algebra in the usual sense (of course, associativity up to bla, etc), or is there a different notion of (higher) Hopf algebras in categories like $\mathsf{2Vect}_{KV}$ which is more suitable in this context? Or both?
