  We present a new parametric lens model for the G165.7+67.0 galaxy cluster,
which was discovered with $Planck$ through its bright submillimeter flux,
originating from a pair of extraordinary dusty star-forming galaxies (DSFGs) at
$z\approx 2.2$. Using JWST and interferometric mm/radio observations, we
characterize the intrinsic physical properties of the DSFGs, which are
separated by only $\sim 1^{\prime\prime}$ (8 kpc) and a velocity difference
$\Delta V \lesssim 600~{\rm km}~{\rm s}^{-1}$ in the source plane, and thus
likely undergoing a major merger. Boasting intrinsic star formation rates ${\rm
SFR}_{\rm IR} = 320 \pm 70$ and $400 \pm 80~ M_\odot~{\rm yr}^{-1}$, stellar
masses ${\rm log}[M_\star/M_\odot] = 10.2 \pm 0.1$ and $10.3 \pm 0.1$, and dust
attenuations $A_V = 1.5 \pm 0.3$ and $1.2 \pm 0.3$, they are remarkably similar
objects. We perform spatially-resolved pixel-by-pixel SED fitting using
rest-frame near-UV to near-IR imaging from JWST/NIRCam for both galaxies,
resolving some stellar structures down to 100 pc scales. Based on their
resolved specific SFRs and $UVJ$ colors, both DSFGs are experiencing
significant galaxy-scale star formation events. If they are indeed interacting
gravitationally, this strong starburst could be the hallmark of gas that has
been disrupted by an initial close passage. In contrast, the host galaxy of the
recently discovered triply-imaged SN H0pe has a much lower SFR than the DSFGs,
and we present evidence for the onset of inside-out quenching and large column
densities of dust even in regions of low specific SFR. Based on the intrinsic
SFRs of the DSFGs inferred from UV through FIR SED modeling, this pair of
objects alone is predicted to yield an observable $1.1 \pm 0.2~{\rm
CCSNe~yr}^{-1}$, making this cluster field ripe for continued monitoring.
