Beware that the forgetful functor on the category of representations of a quasi-triangular Hopf algebra is typically not a braided functor. Rather, the general pattern is that you can reconstruct a bialgebra from a monoidal functor, and then additional structures/properties of the category you started with induces additional structures/properties on that bialgebra. In fact, I claim that $F$ being monoidal plays essentially no role in the sort of conditions you want: once you've identified $B$ with a category of modules for an algebra in $C$, and $F$ with the forgetful functor, then the rest is automatic. Also, in the infinite dimensional setting reconstruction of categories of comodule, rather than modules, usually work better.
That being said, a convenient formalism for that kind of question goes under the name "Barr-Beck monadicity theorem". There are a bunch of versions of those, including version for $\infty$-categories, see e.g. https://ncatlab.org/nlab/show/monadicity+theorem. They give you sufficient condition for $B$ to be the category of modules over a monad on $C$ (there are versions for comonad as well of course).
The trick is that $C$ is a right module over itself and that, in many situation and for an appropriate choice of functors, the map $X \mapsto X \otimes -$ gives a monoidal equivalence
$$C \simeq End_C(C)$$
Where the RHS is the monoidal category of right $C$-linear functors. This means that there is an equivalence of categories between algebra objects in $C$, and right $C$-linear monads on $C$. This is how those monadicity theorems can be applied to your situation.
Edit It's hard giving an answer or a reference without knowing the kind of categories you're interested in, but to be a bit more precise: first you need $B$ to be a $C$-module, and $F$ to be a functor of $C$-module. Now a version of the monadicity theorem, adapted to your situation, tells you that:
- $F$ needs to have a left adjoint $G$ as a $C$-linear functor. In practice this is something you can often check by verifying some conditions on $F$ and $C$. In particular the $C$-linear structure is forced, if there's one compatible with the adjunction.
- $F$ needs to be conservative. If $B$ is abelian, this is implied by $F$ being faithful, and again this is often how it's checked in practice.
- $B$ needs to have, and $F$ needs to preserves, coequalizers of $F$-split pairs. Often something much stronger is true by assumption (e.g. $B$ is closed under finite, or all colimits, and $F$ preserves those).
- you need to know that $FG$, à $C$-linear endofunctor of $C$, is of the form $A \otimes -$ for some object $A \in C$. Again there are many settings in which this is automatic.
Then the unit and the counit of the adjunction make $FG(1_C)=A$ an algebra object in $C$, and $B=A-mod_C$.
Then if $F$ is monoidal, it's well-known that $G$ is automatically oplax monoidal, hence $FG(1_C)=A$ is a coalgebra: this gives you the coproduct. You can then build the R-matrix from the braiding of $B$ and the symmetry of $C$, but there are subtleties in properly defining quasi-triangular bialgebra in arbitrary symmetric monoidal categories, see e.g. https://arxiv.org/abs/math/0604180. However, in concrete cases (e.g. $C=R-mod$ for a commutative ring $R$) this should works the way you'd expect.
