Let G be a braided monoidal groupoid: it does no harm to suppose that the monoidal product on G is strictly associative, so I'll do that.
"With inverses" means that for every object $X$ of G, there is an object $Y$ and an isomorphism $X\otimes Y\approx 1$ in G, where $1$ is the unit object.
I'd like to assume, without loss of generality, that inverses exist "on the nose", so that for every object $X$, there is an object $X^{-1}$ such that $X\otimes X^{-1} = 1$. That is, the objects of G form a group.
First question: am I allowed to do this?
Now I can get a 2-category BG by "delooping" G (using the monoidal structure), so that BG has only one object *, and the category of morphisms BG(*,*) is exactly G. This is a 2-groupoid with one object, and it carries some sort of additional structure encoding the braiding.
A connected 2-groupoid is exactly the same thing as a crossed module, which consists of data $(H,F, d: H\to F, \phi: F\to \mathrm{Aut}(H))$, where $H$ and $F$ are groups and $d$ and $\phi$ are homomorphisms. In terms of G, F is the group of objects of G, while H is the set of 1-morphisms in G with unit object as domain.
Second question: what extra structure do I put on the crossed module to encode the braiding?
I want to understand such G which are free on some set S of objects. In the translation to crossed modules, the group F will have to be the free group on S.
Third question: how do you describe the group H in this crossed module? (That's what I mean by "explicit".)
There's an extensive literature on braided monoidal categories, so I bet someone has thought about this.
(Oh, and I can deloop one more time to get a weak 3-groupoid B2G, and this thing will model a homotopy 3-type X. If G is free, X is a wedge of 2-spheres. Because of this, I know things like the image and kernel of $d: H\to F$. But what is $H$ itself?)