I botched the calculation. While I got the answer I wanted (a quadratic improvement in energy), and while I more-or-less correctly identified the reason for that answer (unintuitive properties of the relativistic velocity addition formula), I did the calculation in the rest frame of one of the particles instead of the zero-momentum rest frame, and thereby obtained a scaling of 1/sqrt(ε) versus 1/ε instead of 1/ε1/4 versus 1/sqrt(ε). As a result, my answer flagrantly violates conservation of energy.
Thanks to rrtucci and perseph0ne. In my defense, I did call it a doofus discovery.