Summary: For part I see [ibid., Paper No. 76, 45 p. (2018; doi:10.1007/JHEP12(2018)076]. The study of form factors has many phenomenologically interesting applications, one of which is Higgs plus gluon amplitudes in QCD. Through effective field theory techniques these are related to form factors of various operators of increasing classical dimension. In this paper we extend our analysis of the first finite top-mass correction, arising from the operator Tr(\(F^3\)), from \(\mathcal{N}=4\) super Yang-Mills to theories with \( \mathcal{N} < 4\), for the case of three gluons and up to two loops. We confirm our earlier result that the maximally transcendental part of the associated Catani remainder is universal and equal to that of the form factor of a protected trilinear operator in the maximally supersymmetric theory. The terms with lower transcendentality deviate from the \( \mathcal{N} =4\) answer by a surprisingly small set of terms involving for example \(\zeta_2\), \(\zeta_3\) and simple powers of logarithms, for which we provide explicit expressions.