I won't give this as an answer because I can't confirm your conditions, but at least it discusses these topics and can be searched: math.tamu.edu/~maguiar/a.pdf

Probably Turaev's book would have this. Briefly, consider the disk with n punctures each labeled with the object $V$ and the boundary labelled by the object $V_i$. The modular functor of the corresponding TQFT assigns to this surface the vector space $\mathrm{Hom}(V_i,V^{\otimes n})$. From the Jones polynomial/Temperley-Lieb algebra perspective these are the irreducible $TL_n(q)$-modules. So the direct sum of these is the minimal faithful module that one can use to define the Jones polynomial (this by $q$-Schur-Weyl duality).

Can this be phrased in the quantum group setting? Something like comparing the $B_n$ representations on $V^{\ot n}$ where $V$ is a $U_q sl_2$ module and the $B_n$ representations from the mapping of $\mathbb{C}B_n$ in $End(V^{\ot n})$? To get a Hilbert space I think you need the root of unity case (and indeed, if you are fixing a level then this is the case). If so, I can give you an answer.

Thanks! I really like Taylor's notes. I will definitely include this as an extra resource if only for the advice in the appendices.

The course is mostly for math majors, but also fulfills a university requirement for a "writing-intensive" course so some science/engineering students take it too (at their own peril!). So it is part structure of proofs and part "math appreciation."

I just might do this after I teach it a couple more times (and get tenure!)

@Peter: Zhenghan Wang's CBMS monograph "Topological Quantum Computation" is now available from the AMS and might fit the bill as a recent survey--particularly chapters 6 and 8 deal with this subject.