how would you prove that polynomial functions are not exponential?

Your proof is correct. You can also say that $\lim_{x\to-\infty}e^x=0$, whereas you have$$\lim_{x\to-\infty}P(x)=\pm\infty$$if $P$ is a non-constant polynomial function. And, clearly, the exponential function is not constant.

Suppose $e^x=P(x)$, where $P$ is a polynomial of degree $n$. Note first that $n\gt0$, since $e^x$ is nonconstant. It follows that $P(2x)$ and $(P(x))^2$ are polynomials of different degrees, namely $n$ and $2n$. But $P(2x)=e^{2x}=(e^x)^2=(P(x))^2$ says they are of the same degree, which is a contradiction. So $e^x$ is not equal to any polynomial.