Chromospheric jets are plausible agents of energy and mass transport in the solar chromosphere, although their driving mechanisms have not yet been elucidated. Magnetic field measurements are key for distinguishing the driving mechanisms of chromospheric jets. We performed a full Stokes synthesis in the infrared range with a realistic radiative magnetohydrodynamics simulation that generated a chromospheric jet to predict spectropolarimetric observations from the Sunrise Chromospheric Infrared spectro-Polarimeter (SCIP) onboard the SUNRISE III balloon telescope. The jet was launched by the collision between the transition region and an upflow driven by the ascending motion of the twisted magnetic field at the envelope of the flux tube. This motion is consistent with upwardly propagating non-linear Alfvénic waves. The upflow could be detected as continuous Doppler signals in the Ca II 849.8 nm line at the envelope where the dark line core intensity and strong linear polarization coexist. The axis of the flux tube was bright in both Fe I 846.8 nm and Ca II 849.8 nm lines with downflowing plasma inside it. The structure, time evolution, and Stokes signals predicted in our study will improve the physical interpretation of future spectropolarimetric observations with SUNRISE III/SCIP.