Any meromorphic modular function of weight $0$ for $\mathrm{SL}(2,\Bbb Z)$ is a rational function of $j$, say $P(j)$. Since your function is holomorphic, $P$ is a polynomial. Since your function has a simple pole at infinity, $P$ has degree one. But $P$ fixes $0$ and $1$, so it is the identity.