This answer only deals with the case that $R$ is infinite. I thought that I would be able to modify it to the finite case - thanks to Ilya Bogdanov for spotting the mistake in my argument. (His answer shows that for finite $R$ such family indeed exists.)
$\newcommand{\mc}[1]{\mathcal{#1}}$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. (Here we're using the assumption that $R$ is infinite.)
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".