Canning lines like the Firestone Walker has do keep the filling area filled with CO2. If Vinny buys what they have or what Pizza Port has, RR will be fine.Oxygenation due to process controls isn't my speciality, but I can easily believe that Vinny is correct and the specifics of how a can/bottle is filled can have a significant effect on how much DO ends up in the product. I would guess that how long it takes between filling and capping matters, but perhaps less than how quickly the can is filled and how much surface area is exposed through turbulence and frothing. So Vinny's concerns that he'll both introduce oxygen and variability by capping four cans at once isn't unfounded.
I'd think some of the best ways to lower the DO would be to fill the cans quickly, from the bottom up, with a filling head designed to minimize turbulent flow, and to cap the cans as quickly as possible after filling. Bonus points if you can keep the whole canning line under a blanket of cold CO2, though that's a hard enough that I imagine that only really big breweries could afford such a thing.