Having difficulty checking for time invariance of discrete system
You did everything correctly. I think that part of your confusion comes from the way you chose the notation. Let me use $v_k[n]$ to denote the system's response to a delayed input $x[n-k]$. $v_k[n]$ is described by the following difference equation:

$$v_k[n]=3v_k^2[n-1]-nx[n-k]+4x[n-k-1]-2x[n-k+1]\tag{1}$$

On the other hand, the delayed response $y[n-k]$ to the original input $x[n]$ is described by

$$y[n-k]=3y^2[n-k-1]-(n-k)x[n-k]+4x[n-k-1]-2x[n-k+1]\tag{2}$$

The system would be time-invariant if $y[n-k]$ and $v_k[n]$ satisfied the same difference equation. From $(1)$ and $(2)$ we can see that that is not the case, hence the system is time-varying. You can see that the multiplicative term $n$ is responsible for making the system time-varying. In general, if there are time-dependent coefficients in a difference equation then the corresponding system is time-varying.