We calculate the lowest even isovector moment of the $F_2$ structure function in $2+1$-flavour lattice QCD with varying quark masses corresponding to $m_\pi \approx [410, 360, 300] \; {\rm MeV}$, at a fixed volume of $V = 48^3 \times 96$ and coupling $\beta = 5.65$ ($a = 0.068(3) \, {\rm fm}$). We directly compute the physical Compton amplitude using the Feynman-Hellmann approach and extract moments of the physical structure function. We report on the quark-mass dependence of the lowest isovector moment and estimate its value at the physical quark-mass point with $\sim 10\%$ uncertainty at fixed $Q^2$. By analysing the $Q^2$ dependence of the moments at the $SU(3)$ symmetric point ($m_\pi \approx 410 \; {\rm MeV}$), we separate the leading- and higher-twist contributions and estimate the parton momentum fraction, $\langle x \rangle_{u-d}$, which agrees with existing results.