This spin structure function of the neutron is traditionally determined by measuring the spin asymmetry of inclusive electron deep-inelastic scattering (DIS) off polarized $^3$He nuclei. In such experiments, correcting for nuclear effects can introduce large systematic uncertainties and model-dependencies. This talk presents our study of the feasibility of suppressing such model dependencies by tagging both spectator protons in the process of DIS off neutrons in $^3$He at the forthcoming Electron-Ion Collider (EIC). This allows us to reconstruct the momentum of the struck neutron to ensure it was nearly on-shell in the initial state, thereby reducing sensitivity to nuclear corrections, and to suppress contributions from electron DIS off protons in $^3$He. We find that the EIC can probe the neutron spin structure from $x_B$ of 0.003 to 0.0651. We further find that the double spectator tagging method results in reduced uncertainties by a factor of 4 on the extracted neutron spin asymmetries over all kinematics, and by a factor of 10 in the low-$x_B$ region, providing high-precision data that will give insight into the spin structure of the nucleon.