Definitely no way (for the calculation to make sense), but this is not a problem. The equality in question is a system of polynomial identities in the Taylor coefficients of $w$, so it is automatically extended to the case when $w$ is just a formal series. (A polynomial which is zero on an open set is zero everywhere.)

Years ago I tried to figure out where this mysterious inversion formulas come from. The result was a paper, An explicit interpolation formula for the Hardy space, A. V. Gavrilov, Siberian Mathematical Journal, volume 41, pages 252–256 (2000). It turned out that there is an interesting analytic side to this linear algebra stuff.

What is usually denoted by $\Gamma(2)$ (at least, in areas I am familiar with) is a subgroup of $PSL_2(\mathbb{Z})$ rather then $SL_2(\mathbb{Z})$. (And yes, it is isomorphic to $F_2$.)

In case anyone is interested, I posted this arxiv.org/abs/2007.12461 .