I would say that the complete delta F term on the right hand side of equation (7) of Note 396(3) must be equated to the complete sum of the binomial expansion. They cannot be equated term by term. However, for v << c Eq. (16) holds, in which delta F is the sum of the fourth and sixth order corrections that you have already worked out, plus higher order terms. The reasoning is that the lagrangian (10) is known to give precession, (UFT377) so if the vacuum correction delta F is equated to the right hand side of Eq. (16), delta F will give the same precession. Therefore precession originates in the vacuum, QED.