Measurements of $CP$ observables in $B\pm \to DK\pm$ and $B\pm \to D\pi \pm$ decays are presented, where $D$ represents a superposition of $D0$ and $D¯0$ states. The $D$ meson is reconstructed in the three-body final state $KS0K\pm \pi \mp$. The analysis uses samples of $B$ mesons produced in proton-proton collisions, corresponding to an integrated luminosity of 1.0, 2.0, and 6.0 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ collected with the LHCb detector at centre-of-mass energies of $\sqrt{s} = $ 7, 8, and 13 TeV, respectively. These measurements are the most precise to date, and provide important input for the determination of the CKM angle $\gamma$.