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.