A 160km altitude cubesat would require a minimum 7.8km/s of delta-v. Anything less than that and it will burn up in the atmosphere after a few orbits. GoFast Rocket achieved a top speed of 1.6 km/s (3580 mph).

It needs to go 4 times faster, and no, a 20% larger rocket won't cut it. 20000% larger by mass would be more like it. It also needs a 2nd stage burning liquid propellant with high specific impulse which is way, way harder to build than a dumb solid propellant rocket.

Noone (not even governments) has been able to send a single stage rocket to orbit.