PARTIAL answer: By a theorem of Solomon (Louis Solomon, Invariants of Finite Reflection Groups, Nagoya Math. J. Volume 22 (1963), pp. 57-64, also Julia Hartmann and Anne V. Shepler, Reflection Groups and Differential Forms), the Noether bound holds when the action of $G$ on $V$ is a reflection group. (Moreover, in this case, even the tensor product $S\left(V\right)\otimes \wedge\left(V\right)$ is generated in degree $\leq \left| G\right|$. I am wondering what can be said about $S\left(V\right)\otimes \wedge\left(W\right)$ for two different reflection representations $V$ and $W$ of $G$.)
This doesn't say anything about the $S_3$-module I am considering to be a potential counterexample, though ($S_3$ doesn't act as a reflection group on it). I would still be very indebted for some Sage code to work out this case.