In fact, every rainbow hypergraph has chromatic number $2$.

Let $H=(V,E)$ be a rainbow hypergraph, $E=\{e_2,e_3,\dots\}$, $|e_n|=n$. Consider a random coloring $c:V\to\{0,1\}$, let $A$ be the event that $c$ is not a proper coloring of $H$, and let $A_n$ be the event that $c$ is constant on $e_n$. Then
$$P(A)=P\left(\bigcup_{n=2}^\infty A_n\right)\lt\sum_{n=2}^\infty P(A_n)=\sum_{n=2}^\infty\frac1{2^{n-1}}=1,$$
so proper $2$-colorings exist and $\chi(H)=2$.