  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 modeling 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) halos and main deflector subhalos. We
apply this machinery to 50 mock lenses --- roughly the number of known quads
--- with warm dark matter (WDM) mass functions exhibiting free-streaming
cutoffs parameterized by the half-mode mass $m_{\rm{hm}}$. Assuming cold dark
matter (CDM), we forecast bounds on $m_{\rm{hm}}$ 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 $2 \sigma$ we constrain $m_{\rm{hm}}<10^{7.9}
\left(10^{8.4}\right) M_{\odot}$, or a 4.4 (3.1) keV thermal relic, with image
flux uncertainties from measurements and lens modeling of $2\%
\left(6\%\right)$. With less severe tidal destruction we constrain
$m_{\rm{hm}}<10^{7} \left(10^{7.4}\right) M_{\odot}$, or an 8.2 (6.2) keV
thermal relic. If dark matter is warm, with $m_{\rm{hm}} = 10^{7.7} M_{\odot}$
(5.1 keV), we would favor WDM with $m_{\rm{hm}} > 10^{7.7} M_{\odot}$ over CDM
with relative likelihoods of 22:1 and 8:1 with flux uncertainties of $2\%$ and
$6\%$, respectively. These bounds improve over those obtained by modeling only
main deflector subhalos because LOS objects produce additional flux
perturbations, especially for high redshift systems. These results indicate
that $\sim 50$ quads can conclusively differentiate between warm and cold dark
matter.
