Actually, its just good old fashioned Galileo-Newtonian relativity that says all inertial frames of reference are equally valid. No Einstein needs to be invoked in this paradox.
In the way of an answer, this is what I can come up with: Particle accelerators operate by taking advantage of the fact (explained by Special Relativity) that mass is convertible into energy and vice versa. So the kinetic energy lost when a formerly rapidly moving particle stops dead in its tracks due to a collision, is free to be used to form the mass of new particles created. What matters is the total kinetic energy of the system, and if in a stationary (laboratory) frame of reference I have two accelerated particles moving toward each other at high speed, that's more total velocity and therefore more total kinetic energy in the system than one accelerated particle moving toward a stationary one at high speed in the lab frame, no matter what frame of reference I switch to to measure the velocities.

The gist is that you are going to get out what you put in, energy wise, and if you put in enough energy to accelerate two beams of particles to near the speed of light you are going to get more energy, and therefore more mass and more new particles out than if you just accelerate one beam of particles to near the speed of light. Even different inertial reference frames moving at different constant velocities will recognize that acceleration has taken place, and therefore that energy has been added to a system. I don't see why momentum has to be invoked, but who am I to argue with Leon Lederman.