Re your last paragraph (v1, "climb the hill"): you have to be careful whether the masses you find are nuclear masses, whose measurement would require a completely-stripped nucleus in a mass spectrometer, or the masses of neutral atoms, including all the electrons and the electronic binding energies. The "mass excesses" I use in my answer are for neutral atoms, and those data make it clear that the reaction $$\rm ^{124}Xe + 54e^- \not\to {}^{124}I + 53e^- + \nu_e$$ is energetically forbidden. —
rob ♦ 20 hours ago