I'm trying to compute the braiding morphisms between simple objects of the so-called metaplectic categories, i.e. the modular categories with fusion rules equivalent to those of $\operatorname{SO}(N)_2 = \operatorname{Rep}(U_q (\mathfrak{so}_N))$ with $q=e^{2\pi i/N}$.
In this survey article about modular categories coming from quantum groups, the following is written:
A complete description of the braiding and associativity maps is quite difficult in general; fortunately one is usually content to know they exist, relying on the $S$-matrix, fusion matrices and twists for most calculations.
My question is, assuming knowledge of the (pre)modular data in the above quote, how does one compute the (traces of) braidings between simple objects? Denoting the twist by $\theta$, the balancing equation $$(\theta_X \otimes \theta_Y)\circ c_{X,Y}\circ c_{Y,X} = \theta_{X\otimes Y}$$ comes to mind, but generally this only lets one recover the double braiding.
For a specific example, I want to compute the braidings of the objects in the adjoint subcategory of $\operatorname{SO}(16)_2$. The fusion rules and quantum dimensions of these objects are present in the AnyonWiki database.