is $F(x,n)=x(-n + 2)$ time invariant?
Your reasoning is correct and we arrive at the same conclusion.

The system's output is given by

$$y[n]=x[-n+2]\tag{1}$$

Let

$$x_2[n]=x[n-k]\tag{2}$$

The corresponding output is

$$y_2[n]=x_2[-n+2]=x[-n-k+2]\tag{3}$$

From $(1)$ the shifted output is

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

Since $(3)$ and $(4)$ are not identical, the system is time-varying.