Perhaps, the form given by [Lagrange inversion theorem](https://en.wikipedia.org/wiki/Lagrange_inversion_theorem) cannot be much simplified here. It expresses the $n$-th coefficient of a series reversion as the sum of $n-1$ values of exponential Bell polynomials. 

From the practical perspective, since $\theta_3-1$ contains nonzero coefficients only at square powers, computation of the $n$-th reversion coefficient amounts to iterating over the partitions of $n-1+k$ into $k$ nonzero squares ([OEIS A243148](https://oeis.org/A243148)) for $k\in\{1,2,\dots,n-1\}$.