rt.representation theory - Is the category of representations of a finite W-algebra monoidal?

This is just to add another point of view to Ben's and David's comments and mainly for your Edit regarding superconformal field theories, as this is more about the affine case than the finite one.

1) There is a formal relation between the finite W-algebra and the affine one: The finite is the Zhu algebra of the affine one (Kac-De Sole '05).

This in particular implies that irreducible modules for the affine algebra are in 1-1 correspondence with irreducible modules for the finite one. From here perhaps your suspicion that if you expect tensor products for modules of the affine ones then you should expect them for the finite one. That article also answer your question regarding a BRST construction of the finite W-algebra (see the appendix, and also the very nice article by Gan-Ginzburg in the finite case)

2) There is a fusion category structure for the affine W-algebra at certain levels, namely, it is proved in some cases when these W-algebras are rational, and it is still conjectured in many others. At any rate, we know of at least a few examples when we indeed have the tensor structure in the categories of modules for the affine W-algebra (See articles by Arakawa and Kac-Wakimoto starting in 2005 up to late '10).

One observation is that the affine W-algebra that is rational is the simple quotient of the affine W-algebra mentioned in 1) above.

3) The functor that you mention is just the "top component" of the homological reduction functor in the affine situation, where you feed a representation for the affine algebra g and you get a representation for the affine W-algebra (the simple one). This functor has been studied by many, there was a conjecture of Frenkel-Kac-Wakimoto regarding the exactness and behaviour of this functor. Arakawa proved (for the principal nilpotent first in Invent. '05 and then in more generality starting in '08, see also Kac-Wakimoto '07) that this functor is exact and it sends irreducible modules to either zero or irreducible modules. He used this to prove existence of modular invariant representations of the W-algebra. In particular, as you mention this gives a way of fusing W-representations by using the fusion of the affine ones.

Actually, Arakawa proved the irreducibility part in the principal nilpotent case, and as far as I know the almost-irreducibility in general (0802.1564). In some cases he can use a previous result in the finite case: this property of sending irreducibles to irreducibles or zero was proved by Brundan-Kleschev in type A (they proved more: that every simple module over W-finite arises in this way).

4) In the cases when you can prove that a) the affine W-algebra is rational, and b) that every module over the affine W-algebra is the Hamiltonian reduction of a module over the affine lie algebra g, then you get fusion for the W-algebra from fusion on g, and I do not see anything wrong with passing to the finite W-algebra by taking Zhu's algebras everywhere, so in this setting I agree with you, and looking at the list in Kac-Wakimoto '07 of possible rational W-algebras, you should get several examples of finite ones. I don't see anything wrong with this but I may be missing something, perhaps some subtlety between the W-algebra and its simple quotient?