Again, ignore cost to LEO - it's not happening that way, this is something we can clearly both be in agreement over. You'd also want it higher most likely. Just the materials cost, let alone design time and assembly would sincerely suggest such a project could not possibly be entertained until first supermassive resources are obtained - like those littering the solar system.
Certainly, it's unlikey to start fabrication tomorrow there's far too much infrastructure that'd need to be in place. Floor lift or stellar harvest. It doesn't hurt none to start working through what would be involved in doing it, however, so when we are better able to do so all the pieces are in place and most if not all problems rectified before anyone actually does anything. Drastically reduces devel costs for such a system.
With regards to making mass more reasonable, a simple electical motor with a rotor arm and counterweight. Spin it up and release it at the "right" point in it's arc and you can "throw" things. And actually put them where you'd intend. It'd possibly not have the range of a 18km oversized railgun, but a series of them could be built with the same or similar mass.