Abstract Strong lensing provides a powerful means of investigating the nature of dark matter as it probes dark matter structure on sub-galactic scales. We present an extension of a forward modelling framework that uses flux ratios from quadruply imaged quasars (quads) to measure the shape and amplitude of the halo mass function, including line-of-sight (LOS) haloes and main deflector subhaloes. We apply this machinery to 50 mock lenses – roughly the number of known quads – with warm dark matter (WDM) mass functions exhibiting free-streaming cut-offs parametrized by the half-mode mass mhm. Assuming cold dark matter (CDM), we forecast bounds on mhm and the corresponding thermal relic particle masses over a range of tidal destruction severity, assuming a particular WDM mass function and mass–concentration relation. With significant tidal destruction, at 2σ we constrain $m_{\rm {hm}}\lt 10^{7.9} \left(10^{8.4}\right) \, \mathrm{M}_{⊙ }$, or a 4.4 (3.1) keV thermal relic, with image flux uncertainties from measurements and lens modelling of $2{{\ \rm per\ cent}} \left(6{{\ \rm per\ cent}}\right)$. With less severe tidal destruction we constrain $m_{\rm {hm}}\lt 10^{7} \left(10^{7.4}\right) \, \mathrm{M}_{⊙ }$, or an 8.2 (6.2) keV thermal relic. If dark matter is warm, with $m_{\rm {hm}} = 10^{7.7} \, \mathrm{M}_{⊙ }$ (5.1 keV), we would favour WDM with $m_{\rm {hm}} \gt 10^{7.7} \, \mathrm{M}_{⊙ }$ over CDM with relative likelihoods of 22:1 and 8:1 with flux uncertainties of $2{{\ \rm per\ cent}}$ and $6{{\ \rm per\ cent}}$, respectively. These bounds improve over those obtained by modelling only main deflector subhaloes because LOS objects produce additional flux perturbations, especially for high-redshift systems. These results indicate that ∼50 quads can conclusively differentiate between WDM and CDM.