Let $1<p<\infty$ and $1<r<\infty$ and let $\mathcal{M}_{r}(g)$ denote $\mathcal{M}(|g|^{r})^{\frac{1}{r}}$, where $\mathcal{M}$ is the Hardy-Littlewood maximal function. Also let $T\in CZO_{\alpha}$. Then I want to show that there exists a constant $C>0$ such that $\forall$ $f\in C^{\infty}_{0}(\mathbb{R}^{n})$ and $b\in BMO(\mathbb{R}^{n})$ we have $$\|[b,T](f)\|_{p}\le C\|b\|_{BMO(\mathbb{R}^{n})}\|f\|_{p}.$$
From the Fefferman-Stein inequality, we have $\|[b,T](f)\|_{p}\le C(n,p)\|\mathcal{M}^{\#}([b,T](f)\|_{p}$.
Note that we have $\mathcal{M}^{\#}([b,T](f))(x)\le C\|b\|_{BMO(\mathbb{R}^{n})}(\mathcal{M}_{r}(Tf)(x)+\mathcal{M}_{r}(f)(x))$, which I have already proven.
Thus $$\begin{aligned}\|[b,T](f)\|_{p}&\le C(n,p)\|\|b\|_{BMO(\mathbb{R}^{n})}(\mathcal{M}_{r}(Tf)+\mathcal{M}_{r}(f))\|_{p}\\&=C'\|b\|_{BMO(\mathbb{R}^{n})}\|\mathcal{M}_{r}(Tf)+\mathcal{M}_{r}(f)\|_{p}\end{aligned}$$
I am stuck here, although I'd guess that I would need to use the boundedness of $\mathcal{M}_{r}$ and $T$ on $L^{p}(\mathbb{R}^{n})$.
You are right exactly and you are almost done. Fix $1 < r < p < \infty$. Note that $\mathcal{M}$ is bounded $L^{p/r} \to L^{p/r}$. We have $$ \| \mathcal{M}_r f \|_p = \| \mathcal{M}(\lvert f \rvert^r) \|_{p/r}^{1/r} \leq C \| \lvert{f}\rvert^r \|_{p/r}^{1/r} = C \| f \|_p $$ so that $\mathcal{M}_r$ is bounded $L^p \to L^p$. Along with boundedness of $T$, this finishes the proof.