I want to suggest a variation on the launch catapult. The rocket assembly is accelerated vertically through momentum transfer, by a reaction driver on the ground. In concept the driver would be a machine gun, or a jet of water. The rocket assembly has a bumper plate (as in Project Orion). Unlike most catapult concepts, there is no track. One resulting problem is the projectiles (or jet) could miss the bumper, and hit the payload (very bad).
Conceptually, this is a low performance ground-mounted rocket engine, and the mass consumption might be an order of magnitude+ greater than the first stage it aspires to replace. High aerodynamic loads are avoided for the rocket assembly, though they certainly affect the jet or projectiles. A distance of 10 km at 2.5g corresponds to 2500 kph (Mach 2). Aiming at that distance is a question; I have no reason to hope a jet of water would stay together, but can't rule it out. Ice composite projectiles might be possible.
Under Project HARP a projectile reached 180 km altitude, much greater than needed here. The necessary rate of fire would be challenging. Projectile launch velocity would need to be modulated, using a magnetic accelerator, to make closing velocity something reasonable.
Perhaps this could get the rocket assembly above the atmosphere, with enough velocity to be useful.