Did you look at prop 5.21 in the paper with Peter? I think that should answer your question.
There are two slightly different questions you could ask. First how does the framing-dependent invariant change. Here it is just (\pm 1)^#L where # is the number of components. Second how does the framing-corrected invariant change? Here it's (\pm 1)^#L (\pm 1)^writhe. In both cases the \pm 1 just measures whether you've changed the FS indicator of your rep V.
If you want to think about things labelled with components labelled by more than one irrep it'll get yuckier.