Let $\alpha = 2^dP(\cos 2\pi /n)$. Then $\alpha$ is a non-zero algebraic integer and thus its norm is a positive integer. The norm is the product of the $\phi(n)$ conjugates of $\alpha$, all of whom are of absolute value at most $n2^d$, so $|\alpha| \ge (1/n2^d)^{\phi(n)-1}$. This is not as good as you'd like, especially for large $n$. I think the problem with fixed $d$ and large $n$ is hard. See the following question:

How small can a sum of a few roots of unity be?

By the way, did you mean the two occurrences of $n$ to be the same?

Let $\alpha = 2^dP(\cos 2\pi /n)$. Then $\alpha$ is a non-zero algebraic integer and thus its norm is a positive integer. The norm is the product of the $\phi(n)$ conjugates of $\alpha$, all of whom are of absolute value at most $n2^d$, so $|\alpha| \ge (1/n2^d)^{\phi(n)-1}$. This is not as good as you'd like, especially for large $n$. I think the problem with fixed $d$ and large $n$ is hard. See the following question:

How small can a sum of a few roots of unity be?

By the way, did you mean the two occurrences of $n$ to be the same?