Are the class numbers of $\mathbb{Q}(\cos(2\pi / m))$ $O(m^n)$ for some fixed $n$?

The answer is no.

By Proposition 2 of Gary Cornell and Michael I. Rosen 's paper The 𝓁-rank of the real class group of cyclotomic fields (paraphased):

Let $L/\mathbb Q$ be an abelian 𝓁-extension of the rationals with Galois group $G$. Assume the inertia group of $2$ is cyclic. If $G$ is the direct sum of $t$ cyclic groups and $s$ finite primes of $\mathbb Q$ ramify in $L$ then the 𝓁-rank of the class group of $L / \mathbb Q \geq \frac{t(t-1)}{2}-s-e$, where $e=0$ for $𝓁$ odd.

Let $A$ be the first $n$ primes with the property that $x^3-2$ has three roots in $\mathbb F_a$ ($a\in A$). By Chebotarev's density theorem, the largest element of $A$ is $O(n \log n)$.

Let $L_a$ be the cubic field with conductor $a$. Such cubic fields must exist, as the property of $a$ ensures that $1$ has three cubic roots in $\mathbb F_a$, so $a=1 \text{ mod }3$. Notice that $2$ splits completely in $L_a$.

Let $L$ be the compositium of all $L_a$ s. Then $L/\mathbb Q$ is an abelian $3$-extension with Galois group $C_3^n$, and $2$ splits completely (so the inertia group of $2$ is trivial). There are only $n$ primes that ramify in $L$, so the size of the class group is at least $\exp (c_1n^2)$ for some $c_1>0$.

Also the conductor is bounded above by $\exp(c_2n\log^2 n)$ for some $c_2$, so the extension $\mathbb Q (2 \cos (2\pi / \prod_{a\in A} a))/L$ has degree at most $\exp(c_2n\log^2 n)$, and thus the class number of $\mathbb Q (2 \cos (2\pi / \prod_{a\in A} a))/L$ is at least $\exp(c_1n^2-c_2n\log^2 n)$. This is asymptotically larger than $\exp(c_2n\log^2 n)^m$ for any $m$.