You might want to check out Hall algebra - it's an algebra on equivalence classes of quiver representations with a product encoded in all possible extensions. It has some geometric meaning as explained, for example, here.

I'm unsure about specifics, but the picture I have in my mind is that one should draw a knot as braid diagram and then stretch them to intersect the boundary. Then to each topological line operator in the bulk one assigns a point vertrex operator in the boundary CFT. The observables then should be some conformal blocks with such insertions.

I would be very cautious about throwing away both zero mode an it's shift. I think if one where to write just $$V_\alpha(z):=:e^{\sum_{n\neq 0} \frac{a_n}{-n}z^{-n}}:$$ they would get something like $$V_\alpha(z)V_\beta(w)=(1-\frac{z}{w})^{\alpha\beta}:V_\alpha(z)V_\beta(w):$$ which is not a behavior you expect from operators in a Lorentz invariant theory.

I'm not sure what might be the meaning of the representation you provide. It's interesting that you can write them exactly in the cases where you get non-zero correlators. But as for whether the currents generate the whole algebra I would say it's not the case even for free Boson, since you can't write operator $:e^{\alpha \phi(z)}:$ in terms of $\partial \phi(z)$ simply because you can't shift zero mode. In general it should be even more problematic, since you don't get any sort of nice formula for vertex operators.