Proving that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic.
$\mathbb{Z}[x]^{\times}=\{\pm 1\}$ and $\mathbb{Q}[x]^{\times}=\mathbb{Q}^{\times}$. This is because $R[x]^{\times}=R^{\times}$ for integral domains $R$.
Although I've not time to read your proof, you could alternatively use that since $\mathbb{Q}$ is a field, $\mathbb{Q}[x]$ is a principal ideal domain whereas in $\mathbb{Z}[x]$ the ideal $(2,x)$ is an example of an ideal that is not principal.
The abelian group underlying $\mathbb Q[Z]$ is divisible while that of $\mathbb Z[X]$ is not, so they are not isomorphic even as abelian groups!