Trouble showing Time Invariance of recursive system
You've actually proved time-invariance already. It's just a matter of clean notation to see this. Let's use $y_1[n]$ to denote the response to a delayed input $x[n-k]$. The sequence $y_1[n]$ satisfies the following difference equation:

$$y_1[n]-4y_1[n-1]+4y_1[n-2]=20x[n-k]+10x[n-k-1]\tag{1}$$

The delayed response to the input $x[n]$ satisfies

$$y[n-k]-4y[n-k-1]+4y[n-k-2]=20x[n-k]+10x[n-k-1]\tag{2}$$

Now we have to check if $y_1[n]$ and $y[n-k]$ satisfy the same difference equation. Comparing $(1)$ and $(2)$ we see that this is indeed the case.