$\newcommand{\mc}[1]{\mathcal{#1}}$Let us assume that $R$ and $\mc M$ fulfill the conditions given in the question. Let $$R=R_1\cup R_2$$ be any decomposition of $R$ into two disjoint infinite subsets.
Clearly, we have $|R_1\cap M|\le 1$ for each $M\in\mc M$. Therefore the system $\mc M\cup\{R_1\}$ is almost disjoint and thus $R_1\in\mc M$ (from maximality).
But this means that the intersection $$R_1\cap R=R$$ is infinite — contradicting the "choosability".