We have investigated hole-carrier-doping effects in a ferromagnetic insulator ${\text{La}}_{2}{\text{MnNiO}}_{6}$. Employing the ab initio band-structure method, we have examined the changes in the electronic structures and the valence states of Sr-doped ${\text{La}}_{2\ensuremath{-}x}{\text{Sr}}_{x}{\text{MnNiO}}_{6}$ with varying Sr doping ratio. On Sr doping, we have found a transition from a ferromagnetic insulating phase to a half-metallic phase. The half-metallic nature in ${\text{La}}_{2\ensuremath{-}x}{\text{Sr}}_{x}{\text{MnNiO}}_{6}$ is found to be robust with respect to the on-site Coulomb correlation of transition metal $3d$ electrons and the antisite disorder at $B$ sites. We have corroborated that the substantially weak x-ray magnetic circular dichroism signal observed for ${\text{La}}_{2\ensuremath{-}x}{\text{Sr}}_{x}{\text{MnNiO}}_{6}$, as compared to the undoped system, is caused by antisite disorder at $B$ sites in a Sr-doped system.