At the planck energy scale, this is no longer the case, the relativistic mass of particles becomes so big that gravitational interaction is too strong to be ignored, and the SM loses it's ability to predict how particles interact, decay, combine etc
So you're guaranteed to then see 'new physics'.
It's not relativistic mass that's the key factor: relativistic mass is frame dependent, and it's not the source of gravity. The relevant factor is stress-energy: energy density, momentum density, pressure, and other stresses. The key factor at the planck energy scale is that the density of stress-energy is high enough that we can no longer have confidence that classical General Relativity is an accurate description of gravity; we expect to see quantum gravity phenomena at that stress-energy density.
The stress-energy tensor does have a momentum component, but remember that all components of a tensor are frame-dependent. So is "gravitational pull". Obviously the trajectories of two particles passing each other at relativistic speeds will be different from the trajectories of two particles at are at rest relative to each other at some instant; but the difference is not quite as simple as "more gravitational pull", although the two particles having relativistic velocities does mean that the center of mass energy of the system is larger than it would be if both particles started out at rest.
(Actually, the electromagnetic interaction between electrons is so much stronger than the gravitational that the gravitational effects are negligible in the scenario as you state it; but we could eliminate that issue by considering, say, two neutrons instead. My comments above assume that the scenario has been modified accordingly, which is why I said "particles" instead of "electrons".)