  We report new radio observations of SDSS J090122.37+181432.3, a strongly
lensed star-forming galaxy at $z=2.26$. We image 1.4 GHz (L-band) and 3 GHz
(S-band) continuum using the VLA and 1.2 mm (band 6) continuum with ALMA, in
addition to the CO(7-6) and CI(${\rm ^3P_2\rightarrow ^3\!P_1}$) lines, all at
$\lesssim1.^{\prime\prime}7$ resolution. Based on the VLA integrated flux
densities, we decompose the radio spectrum into its free-free (FF) and
non-thermal components. The infrared-radio correlation (IRRC) parameter $q_{\rm
TIR}=2.65_{-0.31}^{+0.24}$ is consistent with expectations for star forming
galaxies. We obtain radio continuum-derived SFRs that are free of dust
extinction, finding $\rm {620}_{-220}^{+280}\,M_\odot\,yr^{-1}$, $\rm
{230}_{-160}^{+570}\,M_\odot\,yr^{-1}$, and $\rm
{280}_{-120}^{+460}\,M_\odot\,yr^{-1}$ from the FF emission, non-thermal
emission, and when accounting for both emission processes, respectively, in
agreement with previous results. We estimate the gas mass from the CI(${\rm
^3P_2\rightarrow ^3\!P_1}$) line as $M_{\rm
gas}=(1.2\pm0.2)\times10^{11}\,M_\odot$, which is consistent with prior
CO(1-0)-derived gas masses. Using our new IR and radio continuum data to map
the SFR, we assess the dependence of the Schmidt-Kennicutt relation on choices
of SFR and gas tracer for $\sim{\rm kpc}$ scales. The different SFR tracers
yield different slopes, with the IR being the steepest, potentially due to
highly obscured star formation in J0901. The radio continuum maps have the
lowest slopes and overall fidelity for mapping the SFR, despite producing
consistent total SFRs. We also find that the Schmidt-Kennicutt relation slope
is flattest when using CO(7-6) or CI(${\rm ^3P_2\rightarrow ^3\!P_1}$) to trace
gas mass, suggesting that those transitions are not suitable for tracing the
bulk molecular gas in galaxies like J0901.
