I wonder how much it would help to have multiple projectiles firing in close succession with the payload last. Centrifuges seem like an ideal launcher for such a strategy, since you could finely tune the timing of successive releases.

If you can do that, and design a booster to round out the orbit that can withstand the acceleration you're golden. You have a system for cheaply delivering raw materials to LEO. Does a block of aluminum have sufficient tensile strength to be spun up to the required energy at reasonable diameter?