  We study a population of significantly sub-solar enrichment galaxies at
z=1.99, to investigate how molecular gas, dust and star-formation relate in
low-metallicity galaxies at the peak epoch of star-formation. We target our
sample with several deep ALMA and VLA datasets, and find no individual
detections of CO[4-3], CO[1-0] or dust, in stark contrast to the $>$60%
detection rate expected for solar-enrichment galaxies with these MS Halpha
SFRs. We find that both low and high density molecular gas (traced by CO[1-0]
and CO[4-3] respectively) are affected by the low enrichment, showing sample
average (stacked) luminosity deficits $>$0.5-0.7 dex below expectations. This
is particularly pertinent for the use of high-J CO emission as a proxy of
instantaneous star-formation rate. Our individual galaxy data and stacked
constraints point to a strong inverse dependence $\propto Z^{\gamma}$ of
gas-to-dust ratios (G/D) and CO-to-H$_{2}$ conversion factors (aco) on
metallicity at z$\sim$2, with $\gamma_{\rm G/D}<$-2.2 and $\gamma_{\alpha_{\rm
CO}}<$-0.8, respectively. We quantify the importance of comparing G/D and aco
vs. metallicity trends from the literature on a common, suitably normalised
metallicity scale. When accounting for systematic offsets between different
metallicity scales, our z$\sim$2 constraints on these scaling relations are
consistent with the corresponding relations for local galaxies. However, among
those local relations, we favour those with a steep/double power-law dependence
of G/D on metallicity. Finally, we discuss the implications of these findings
for (a) gas mass measurements for sub-M$^{*}$ galaxies, and (b) efforts to
identify the characteristic galaxy mass scale contributing most to the comoving
molecular gas density at z=2.
