In (5.7.1)-(5.7.15), Weinberg derives the form of a massive covariant field operator of any spin in terms of Clebsch-Gordan coefficients, but the English version available to me uses quite different notation. The relation $F^\sigma=\pm G^{-\sigma}$ mentioned in the question is an intermediate result of that calculation, based on the Lorentz covariance properties form Section 5.1.

The relation possibly follows more directly from the causality requirement used by Weinberg later to deduce (5.7.27). Just assuming the first formula above and requiring the commutator at spacelike arguments to vanish gives relations between $F$ and $G$ that must be satisfied. They should be strong enough to force the above relation and the connection between spin and statistics, but I didn't do the calculations. Treating the equal time case might be enough.

But this causality argument is unlikely to give the explicit form (5.7/14/15) of $F$ and $G$, so the more detailed calculations of Weinberg don't become superfluous.